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ЦТ Mathematics practice questions

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Question 1Numbers and computation: natural, integer, rational, irrational and real numbers; powers and roots; proportions and percentages

The price of an item is first increased by 20%, and then the new price is decreased by 20%. What is the net effect on the price?

  1. a 40% decrease
  2. a 4% decrease Correct answer.
  3. no net change
  4. a 4% increase

Explanation

Multiplying by 1.20 and then by 0.80 gives 1.20 × 0.80 = 0.96, which is 96% of the original price. The price therefore falls by 4%. The two percentages are taken from different amounts, so they do not cancel.

Question 2Numbers and computation: natural, integer, rational, irrational and real numbers; powers and roots; proportions and percentages

Which of the following numbers is irrational?

  1. √2 Correct answer.
  2. 22/7
  3. 0.75
  4. −5

Explanation

An irrational number cannot be expressed as a quotient of two integers, and its decimal expansion neither terminates nor repeats. The square root of 2 has such an expansion, so it is irrational, while 0.75, 22/7 and −5 can all be written as fractions of integers.

Question 3Numbers and computation: natural, integer, rational, irrational and real numbers; powers and roots; proportions and percentages

Simplify the expression (2^5 × 2^(−3)) ÷ 2^(−6).

  1. 1/4
  2. 4
  3. 1/256
  4. 256 Correct answer.

Explanation

In the numerator the exponents add: 2^5 × 2^(−3) = 2^2. Dividing by 2^(−6) subtracts the exponent, giving 2^(2 − (−6)) = 2^8 = 256.

Question 4Numbers and computation: natural, integer, rational, irrational and real numbers; powers and roots; proportions and percentages

Calculate √50 × √18.

  1. 15√2
  2. 68
  3. 30 Correct answer.
  4. √68

Explanation

The product of square roots equals the square root of the product: √50 × √18 = √(50 × 18) = √900 = 30. Equivalently, 5√2 × 3√2 = 15 × 2 = 30.

Question 5Numbers and computation: natural, integer, rational, irrational and real numbers; powers and roots; proportions and percentages

Which of these numbers is divisible by 3 but not divisible by 9?

  1. 1539
  2. 5214 Correct answer.
  3. 3402
  4. 2718

Explanation

A number is divisible by 3 exactly when its digit sum is a multiple of 3, and by 9 when that sum is a multiple of 9. For 5214 the digit sum is 5 + 2 + 1 + 4 = 12, which is divisible by 3 but not by 9.

Question 6Numbers and computation: natural, integer, rational, irrational and real numbers; powers and roots; proportions and percentages

Find the value of (2/3 − 1/4) ÷ 5/6.

  1. 5/6
  2. 1/4
  3. 5/12
  4. 1/2 Correct answer.

Explanation

Subtract first using the common denominator 12: 2/3 − 1/4 = 8/12 − 3/12 = 5/12. Dividing by 5/6 means multiplying by its reciprocal 6/5, so the result is 5/12 × 6/5 = 6/12 = 1/2.

Question 7Numbers and computation: natural, integer, rational, irrational and real numbers; powers and roots; proportions and percentages

The repeating decimal 0.121212… is equal to which fraction?

  1. 2/17
  2. 4/33 Correct answer.
  3. 12/100
  4. 1/12

Explanation

Let x = 0.121212…; then 100x = 12.121212…, and 100x − x = 99x = 12. Hence x = 12/99 = 4/33, which has exactly the required repeating expansion.

Question 8Numbers and computation: natural, integer, rational, irrational and real numbers; powers and roots; proportions and percentages

A sum of 420 is divided among three people in the ratio 2 : 3 : 5. What is the largest share?

  1. 210 Correct answer.
  2. 200
  3. 84
  4. 126

Explanation

The ratio contains 2 + 3 + 5 = 10 equal parts, so one part is 420 ÷ 10 = 42. The largest share is 5 parts, namely 5 × 42 = 210.

Question 9Numbers and computation: natural, integer, rational, irrational and real numbers; powers and roots; proportions and percentages

Simplify (√3 + √2)² − (√3 − √2)².

  1. 10
  2. 2√6
  3. 4√6 Correct answer.
  4. 2

Explanation

Expanding each square gives (√3 + √2)² = 3 + 2√6 + 2 = 5 + 2√6 and (√3 − √2)² = 3 − 2√6 + 2 = 5 − 2√6. Subtracting, (5 + 2√6) − (5 − 2√6) = 4√6.

Question 10Numbers and computation: natural, integer, rational, irrational and real numbers; powers and roots; proportions and percentages

Money is invested at 10% per year, with the interest added to the capital at the end of each year. After how many complete years does the amount first exceed 1.5 times the original capital?

  1. 6
  2. 3
  3. 4
  4. 5 Correct answer.

Explanation

Each year the capital is multiplied by 1.1, so after n years it equals 1.1^n times the original. Now 1.1^4 = 1.4641, which is still below 1.5, while 1.1^5 = 1.61051, which is above 1.5; therefore five complete years are needed.

Question 11Numbers and computation: natural, integer, rational, irrational and real numbers; powers and roots; proportions and percentages

How many natural divisors does the number 72 have?

  1. 12 Correct answer.
  2. 8
  3. 10
  4. 14

Explanation

Write 72 in prime factorisation: 72 = 2³ · 3². A divisor is formed by choosing an exponent of 2 from 0 to 3 (four choices) and an exponent of 3 from 0 to 2 (three choices). Multiplying the numbers of choices gives 4 · 3 = 12 divisors.

Question 12Numbers and computation: natural, integer, rational, irrational and real numbers; powers and roots; proportions and percentages

Find the value of |−7| + (−3)² − 5.

  1. 21
  2. −3
  3. 11 Correct answer.
  4. 5

Explanation

The absolute value of −7 is 7, and (−3)² = 9 because squaring a negative number gives a positive result. Substituting gives 7 + 9 − 5 = 11.

Question 13Numbers and computation: natural, integer, rational, irrational and real numbers; powers and roots; proportions and percentages

18% of a certain number is equal to 54. Find this number.

  1. 972
  2. 324
  3. 300 Correct answer.
  4. 270

Explanation

If 18% of the number is 54, then 1% of it is 54 ÷ 18 = 3, so the whole number (100%) is 3 · 100 = 300. The same result comes from dividing 54 by 0,18.

Question 14Numbers and computation: natural, integer, rational, irrational and real numbers; powers and roots; proportions and percentages

Find the value of 27^(2/3) − 16^(1/4).

  1. 7 Correct answer.
  2. 9
  3. 5
  4. 11

Explanation

The exponent 2/3 means the cube root of 27 squared: the cube root is 3, and 3² = 9. The exponent 1/4 means the fourth root of 16, which is 2. Therefore the difference is 9 − 2 = 7.

Question 15Numbers and computation: natural, integer, rational, irrational and real numbers; powers and roots; proportions and percentages

Which of the following expressions has a rational value?

  1. √2 + √7
  2. √3 + √12
  3. √2 · √8 Correct answer.
  4. √5 · √10

Explanation

The product √2 · √8 equals √16 = 4, and 4 can be written as the fraction 4/1, so it is rational. The other expressions simplify to 3√3, 5√2 and √2 + √7, none of which can be written as a ratio of integers.

Question 16Numbers and computation: natural, integer, rational, irrational and real numbers; powers and roots; proportions and percentages

Which inequality correctly orders the numbers 2√3, 3√2 and 5?

  1. 5 < 2√3 < 3√2
  2. 2√3 < 3√2 < 5 Correct answer.
  3. 2√3 < 5 < 3√2
  4. 3√2 < 2√3 < 5

Explanation

All three numbers are positive, so squaring preserves the order: (2√3)² = 12, (3√2)² = 18 and 5² = 25. Since 12 < 18 < 25, we get 2√3 < 3√2 < 5.

Question 17Numbers and computation: natural, integer, rational, irrational and real numbers; powers and roots; proportions and percentages

A team of 8 workers can finish a job in 15 days. After 5 days of work, 3 workers leave. How many more days do the remaining workers need to finish the job, working at the same rate?

  1. 20
  2. 14
  3. 16 Correct answer.
  4. 18

Explanation

The whole job equals 8 · 15 = 120 worker-days. In the first 5 days the team completes 8 · 5 = 40 worker-days, leaving 80 worker-days. With 5 workers remaining, the time needed is 80 ÷ 5 = 16 days.

Question 18Numbers and computation: natural, integer, rational, irrational and real numbers; powers and roots; proportions and percentages

In a school, 60% of the students are girls. Among the girls, 25% wear glasses, and among the boys, 20% wear glasses. What percentage of all the students wear glasses?

  1. 24%
  2. 21%
  3. 22,5%
  4. 23% Correct answer.

Explanation

Girls are 60% of the students and 25% of them wear glasses, which gives 0,60 · 0,25 = 0,15 of all students. Boys are 40% and 0,40 · 0,20 = 0,08 of all students wear glasses. The total is 0,15 + 0,08 = 0,23, that is 23%.

Question 19Numbers and computation: natural, integer, rational, irrational and real numbers; powers and roots; proportions and percentages

Simplify the expression √(7 + 4√3), given that the result is positive.

  1. 1 + 2√3
  2. 3 + √3
  3. 2√2 + 1
  4. 2 + √3 Correct answer.

Explanation

Look for numbers a and b with a² + b² = 7 and 2ab = 4√3, so ab = 2√3. Choosing a = 2 and b = √3 satisfies both conditions, since 4 + 3 = 7 and 2 · 2 · √3 = 4√3. Hence √(7 + 4√3) = 2 + √3.

Question 20Numbers and computation: natural, integer, rational, irrational and real numbers; powers and roots; proportions and percentages

Which of the numbers 2^30, 3^20, 5^12 and 6^10 is the greatest?

  1. 3^20 Correct answer.
  2. 5^12
  3. 6^10
  4. 2^30

Explanation

Rewrite 2^30 = (2³)^10 = 8^10 and 3^20 = (3²)^10 = 9^10, so 3^20 exceeds 2^30, and 6^10 < 3^10 · 3^10 = 3^20. Also 3^20 = (3^5)^4 = 243^4 while 5^12 = (5³)^4 = 125^4, so 3^20 is the largest.

Question 21Algebraic expressions, equations, systems of equations and identities

Simplify the expression 3(2x - 5) - 2(x - 4).

  1. 8x - 7
  2. 4x + 7
  3. 4x - 23
  4. 4x - 7 Correct answer.

Explanation

First distribute each factor: 3(2x - 5) = 6x - 15 and -2(x - 4) = -2x + 8. Combining like terms gives 6x - 2x - 15 + 8 = 4x - 7. A sign slip in the second bracket usually produces the distractor 4x - 23.

Question 22Algebraic expressions, equations, systems of equations and identities

Factor the expression 9a^2 - 16b^2 completely over the integers.

  1. (3a - 2b)(3a + 2b)
  2. (3a - 4b)^2
  3. (9a - 4b)(a + 4b)
  4. (3a - 4b)(3a + 4b) Correct answer.

Explanation

A difference of squares a^2 - b^2 factors as (a - b)(a + b). Here 9a^2 = (3a)^2 and 16b^2 = (4b)^2, so the factors are (3a - 4b)(3a + 4b). Expanding the result returns the original binomial with no middle term.

Question 23Algebraic expressions, equations, systems of equations and identities

Solve the equation (x + 3)/4 - (x - 1)/6 = 1.

  1. 1 Correct answer.
  2. 5
  3. -1
  4. 11

Explanation

Multiply both sides by the common denominator 12: 3(x + 3) - 2(x - 1) = 12. Expanding gives 3x + 9 - 2x + 2 = 12, so x + 11 = 12 and x = 1. Substituting x = 1 back into the original equation confirms the result.

Question 24Algebraic expressions, equations, systems of equations and identities

Solve the system of equations: 3x + 2y = 16 and x - y = 2.

  1. (1; 3)
  2. (4; -2)
  3. (4; 2) Correct answer.
  4. (2; 4)

Explanation

The second equation gives x = y + 2. Substituting into the first yields 3(y + 2) + 2y = 16, so 5y = 10 and y = 2, hence x = 4. Checking (4; 2) in both equations gives 12 + 4 = 16 and 4 - 2 = 2, so the pair is correct.

Question 25Algebraic expressions, equations, systems of equations and identities

The numbers x1 and x2 are the roots of the equation x^2 - 5x + 3 = 0. Find x1^2 + x2^2.

  1. 19 Correct answer.
  2. 22
  3. 25
  4. 31

Explanation

By Vieta's formulas, x1 + x2 = 5 and x1*x2 = 3. Using the identity x1^2 + x2^2 = (x1 + x2)^2 - 2*x1*x2 gives 25 - 6 = 19.

Question 26Algebraic expressions, equations, systems of equations and identities

Simplify the expression (x^2 - 9)/(x^2 + x - 12) for all values of x where it is defined.

  1. (x + 3)/(x + 4) Correct answer.
  2. (x + 3)/(x - 4)
  3. (x - 3)/(x + 4)
  4. (x - 3)/(x - 4)

Explanation

Factor both parts: x^2 - 9 = (x - 3)(x + 3) and x^2 + x - 12 = (x + 4)(x - 3). Cancelling the common factor (x - 3), which is allowed for x not equal to 3, leaves (x + 3)/(x + 4).

Question 27Algebraic expressions, equations, systems of equations and identities

Find the numbers a and b for which x^2 + 6x + 11 = (x + a)^2 + b holds for every real x.

  1. a = 3, b = 2 Correct answer.
  2. a = 6, b = 2
  3. a = 3, b = 11
  4. a = -3, b = 2

Explanation

Expanding the right side gives (x + a)^2 + b = x^2 + 2ax + a^2 + b. Comparing coefficients with x^2 + 6x + 11 gives 2a = 6, so a = 3, and then a^2 + b = 11, so b = 2.

Question 28Algebraic expressions, equations, systems of equations and identities

Solve the equation sqrt(x + 6) = x over the real numbers.

  1. 3 Correct answer.
  2. -2
  3. no real solutions
  4. 3 and -2

Explanation

Squaring both sides gives x + 6 = x^2, that is x^2 - x - 6 = 0, whose roots are 3 and -2. The square root is always non-negative, so the right side x must also be non-negative, which makes -2 an extraneous root. Only x = 3 remains, and it checks: sqrt(9) = 3.

Question 29Algebraic expressions, equations, systems of equations and identities

The real numbers x and y satisfy x + y = 7 and xy = 10. Find the value of x^3 + y^3.

  1. 91
  2. 105
  3. 133 Correct answer.
  4. 343

Explanation

Use the symmetric identity x^3 + y^3 = (x + y)^3 - 3xy(x + y). Substituting x + y = 7 and xy = 10 gives 343 - 3*10*7 = 343 - 210 = 133. The same value follows by first finding x = 5 and y = 2 from the quadratic t^2 - 7t + 10 = 0.

Question 30Algebraic expressions, equations, systems of equations and identities

The quadratic equation x^2 + (a - 2)x + (a + 1) = 0 in x has two equal real roots. Find all possible values of the parameter a.

  1. 8
  2. -8 and 0
  3. 0 and 8 Correct answer.
  4. 0

Explanation

A quadratic has a repeated root exactly when its discriminant is zero. Here D = (a - 2)^2 - 4(a + 1) = a^2 - 8a = a(a - 8), which vanishes at a = 0 and a = 8. Both values turn the trinomial into a perfect square, so both must be included in the answer.

Question 31Algebraic expressions, equations, systems of equations and identities

Expand and simplify (x + 5)(x - 2).

  1. x^2 + 7x - 10
  2. x^2 - 3x - 10
  3. x^2 + 3x - 10 Correct answer.
  4. x^2 + 3x + 10

Explanation

Multiply each term in the first bracket by each term in the second: x·x + x·(-2) + 5·x + 5·(-2). This gives x^2 - 2x + 5x - 10, and collecting like terms yields x^2 + 3x - 10.

Question 32Algebraic expressions, equations, systems of equations and identities

Solve the equation 5 - 2x = 11.

  1. x = -8
  2. x = -3 Correct answer.
  3. x = 8
  4. x = 3

Explanation

Subtract 5 from both sides to obtain -2x = 6, then divide by -2. This gives x = -3; checking: 5 - 2(-3) = 5 + 6 = 11.

Question 33Algebraic expressions, equations, systems of equations and identities

Solve the system of equations: x + y = 5 and x - y = 1.

  1. (2, 3)
  2. (3, 2) Correct answer.
  3. (4, 1)
  4. (1, 4)

Explanation

Adding the two equations eliminates y: (x + y) + (x - y) = 5 + 1, so 2x = 6 and x = 3. Substituting into x + y = 5 gives y = 2, so the solution is (3, 2).

Question 34Algebraic expressions, equations, systems of equations and identities

Simplify (x^2 - 4x)/(x^2 - 16) for all values of x where the expression is defined.

  1. x/(x + 4) Correct answer.
  2. x/(x - 4)
  3. (x - 4)/(x + 4)
  4. (x + 4)/x

Explanation

Factor the numerator as x(x - 4) and the denominator as (x - 4)(x + 4). Cancelling the common factor x - 4 gives x/(x + 4), valid for x not equal to 4 or -4.

Question 35Algebraic expressions, equations, systems of equations and identities

The roots of x^2 - 7x + 4 = 0 are p and q. Find 1/p + 1/q.

  1. 4/7
  2. 1/4
  3. 7
  4. 7/4 Correct answer.

Explanation

By Vieta's formulas, p + q = 7 and pq = 4. Then 1/p + 1/q = (p + q)/(pq) = 7/4.

Question 36Algebraic expressions, equations, systems of equations and identities

Express 2x^2 - 12x + 7 in the form a(x - h)^2 + k.

  1. 2(x - 3)^2 + 7
  2. 2(x - 3)^2 - 11 Correct answer.
  3. 2(x + 3)^2 - 11
  4. 2(x - 6)^2 - 29

Explanation

Factor 2 from the first two terms: 2(x^2 - 6x) + 7. Since x^2 - 6x = (x - 3)^2 - 9, the expression becomes 2[(x - 3)^2 - 9] + 7 = 2(x - 3)^2 - 11.

Question 37Algebraic expressions, equations, systems of equations and identities

Solve sqrt(2x + 5) = x - 1 over the real numbers.

  1. -1
  2. 4
  3. 2 - 2√2
  4. 2 + 2√2 Correct answer.

Explanation

Squaring both sides gives 2x + 5 = (x - 1)^2 = x^2 - 2x + 1, so x^2 - 4x - 4 = 0. The quadratic formula gives x = 2 ± 2√2; the condition x - 1 ≥ 0 rejects 2 - 2√2, leaving x = 2 + 2√2.

Question 38Algebraic expressions, equations, systems of equations and identities

Factor x^3 - 2x^2 - 9x + 18 completely.

  1. (x - 2)(x - 3)^2
  2. (x + 2)(x - 3)(x + 3)
  3. (x - 2)(x - 3)(x + 3) Correct answer.
  4. (x - 2)(x + 3)^2

Explanation

Group terms as x^2(x - 2) - 9(x - 2), giving (x - 2)(x^2 - 9). The difference of squares x^2 - 9 factors as (x - 3)(x + 3), so the full factorization is (x - 2)(x - 3)(x + 3).

Question 39Algebraic expressions, equations, systems of equations and identities

Find all real solutions of |x^2 - 4x| = 3.

  1. 1, 3
  2. 1, 3, 2 ± √7 Correct answer.
  3. 2 ± √7
  4. -1, 1, 3, 2 - √7, 2 + √7, -2 + √7

Explanation

The absolute value equation splits into x^2 - 4x = 3 and x^2 - 4x = -3. The first gives x = 2 ± √7; the second gives x^2 - 4x + 3 = 0, so x = 1 or x = 3. All four values satisfy the original equation.

Question 40Algebraic expressions, equations, systems of equations and identities

Solve x^2 + 1/x^2 = 7, where x ≠ 0.

  1. 3, -3, 1, -1, (7 + 3√5)/2, (7 - 3√5)/2, -(7 + 3√5)/2, -(7 - 3√5)/2
  2. (3 + √5)/2 and (3 - √5)/2 only
  3. (7 + 3√5)/2 and (7 - 3√5)/2 only
  4. ±(3 + √5)/2 and ±(3 - √5)/2 Correct answer.

Explanation

Multiply by x^2 to get x^4 - 7x^2 + 1 = 0. Setting y = x^2 gives y = (7 ± 3√5)/2; taking square roots and simplifying yields x = ±(3 + √5)/2 and x = ±(3 - √5)/2.

Question 41Algebraic expressions, equations, systems of equations and identities

Simplify the expression (2a^2b)^3 · (-3ab^2), assuming a and b are nonzero.

  1. -6a^7b^5
  2. 24a^7b^5
  3. -24a^7b^5 Correct answer.
  4. -24a^6b^6

Explanation

Cube each factor: (2a^2b)^3 = 8a^6b^3. Multiplying by -3ab^2 gives -24a^7b^5, since the exponents add: a has 6+1=7 and b has 3+2=5.

Question 42Algebraic expressions, equations, systems of equations and identities

Solve the equation (2x - 1)/3 = (x + 4)/2 for real x.

  1. 10
  2. 14 Correct answer.
  3. -14
  4. 2

Explanation

Cross-multiplication gives 2(2x-1)=3(x+4), so 4x-2=3x+12. Subtracting 3x and adding 2 yields x=14, and substitution confirms that this value satisfies the original equation.

Question 43Algebraic expressions, equations, systems of equations and identities

If p = -2 and q = 3, what is the value of p^2q - pq^2?

  1. 6
  2. -6
  3. 30 Correct answer.
  4. -30

Explanation

Substitute p=-2 and q=3: p^2q=4·3=12 and pq^2=(-2)·9=-18. Then p^2q - pq^2 = 12 - (-18) = 30, because subtracting a negative adds its absolute value.

Question 44Algebraic expressions, equations, systems of equations and identities

Solve the equation sqrt(3x - 2) = x - 2 over the real numbers.

  1. 1 and 6
  2. 2
  3. 6 Correct answer.
  4. 1

Explanation

Squaring gives 3x-2=(x-2)^2, so x^2-7x+6=0 and x=1 or x=6. The original equation requires x-2≥0, so x=1 is rejected and only x=6 remains.

Question 45Algebraic expressions, equations, systems of equations and identities

The real numbers x and y satisfy x + y = 6 and x^2 + y^2 = 20. Find the value of xy.

  1. 28
  2. 8 Correct answer.
  3. 16
  4. 4

Explanation

Use (x+y)^2 = x^2+y^2+2xy. Substituting gives 36=20+2xy, so 2xy=16 and xy=8. This also matches the pair 2 and 4, whose sum is 6 and squares sum to 20.

Question 46Algebraic expressions, equations, systems of equations and identities

The roots of x^2 - 6x + 4 = 0 are r and s. Find r^2s + rs^2.

  1. 24 Correct answer.
  2. 10
  3. 6
  4. -24

Explanation

By Vieta's formulas, r+s=6 and rs=4. Factoring r^2s+rs^2 gives rs(r+s), so the value is 4·6=24.

Question 47Algebraic expressions, equations, systems of equations and identities

Solve the equation 1/(x - 1) + 1/(x + 1) = 3/(x^2 - 1) for real x.

  1. x = -1/2
  2. x = 0
  3. x = 1/2
  4. x = 3/2 Correct answer.

Explanation

The domain excludes x=1 and x=-1. Combining the left side gives 2x/(x^2-1), so 2x=3 and x=3/2, which lies in the domain.

Question 48Algebraic expressions, equations, systems of equations and identities

The polynomial P(x)=x^3 + kx^2 - 5x + 6 is divisible by x - 2. Find k.

  1. -2
  2. -1 Correct answer.
  3. 1
  4. 2

Explanation

By the remainder theorem, divisibility by x-2 requires P(2)=0. Substituting gives 8+4k-10+6=4k+4, so 4k+4=0 and k=-1.

Question 49Algebraic expressions, equations, systems of equations and identities

Find all real solutions of x^2 + 4x + 5 = 2/(x + 2).

  1. x = -1 Correct answer.
  2. x = -2
  3. x = 1
  4. no real solutions

Explanation

Let t=x+2. Then x^2+4x+5=t^2+1, and the equation becomes t^2+1=2/t, or t^3+t-2=0. Since t=1 is a root and t^2+t+2 has negative discriminant, t=1 is the only real solution, so x=-1.

Question 50Algebraic expressions, equations, systems of equations and identities

Which expression is identically equal to (a - b)^2 + 4ab for all real numbers a and b?

  1. (a - b)^2
  2. (a + b)^2 Correct answer.
  3. a^2 + b^2
  4. (a + 2b)^2

Explanation

Expand the square: (a - b)^2 = a^2 - 2ab + b^2. Adding 4ab gives a^2 + 2ab + b^2, which is the expansion of (a + b)^2.