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Collect like terms: the x-terms give 5x − 2x = 3x and the y-terms give 3y + 7y = 10y. Only terms with identical variable parts may be combined, so the simplified expression is 3x + 10y.
The distributive law gives 4 × 2a = 8a and 4 × (−3) = −12, so the expansion is 8a − 12 + 5a. Collecting like terms gives 13a − 12; a final answer should be fully simplified with no repeated variable terms.
Subtract 3x from both sides to get 4x − 4 = 20, then add 4 to both sides to get 4x = 24. Dividing both sides by 4 gives x = 6. Each step must be applied to the whole of both sides to keep the equation balanced.
Divide both sides by π to obtain r² = A/π, then take the positive square root because r is a length. This gives r = √(A/π). Rearranging means undoing each operation in reverse order.
The cost has a fixed part that does not depend on time and a variable part that grows at $55 per hour. So C = 55h + 80, where 55 is the gradient (rate per hour) and 80 is the vertical intercept (fixed fee).
Gradient equals the change in y divided by the change in x. Here the rise is 13 − 5 = 8 and the run is 6 − 2 = 4, so the gradient is 8 ÷ 4 = 2.
Multiply the coefficients: 3 × 4 = 12. When multiplying powers with the same base, add the indices: x² × x³ = x⁵. The simplified product is therefore 12x⁵.
Direct variation means y = kx for some constant k. Substituting gives 18 = 6k, so k = 3, and the rule is y = 3x. When x = 10, y = 3 × 10 = 30.
Adding the two equations eliminates y because y + (−y) = 0, giving 3x = 12, so x = 4. Substituting into x − y = 1 gives 4 − y = 1, so y = 3. Checking in 2x + y gives 8 + 3 = 11, which is correct.
We need two numbers that multiply to −24 and add to −5; these are −8 and 3. So the factorisation is (x − 8)(x + 3). Expanding back gives x² + 3x − 8x − 24 = x² − 5x − 24, which confirms the result.
Here a = 1, b = 2 and c = −4, so the discriminant is b² − 4ac = 4 + 16 = 20. Then x = (−2 ± √20)/2 = (−2 ± 4.472)/2, giving x ≈ 1.24 or x ≈ −3.24.
Subtract 3 from both sides to get −2x < 8. Dividing both sides by −2 reverses the inequality sign, giving x > −4. Reversing the sign is essential whenever an inequality is multiplied or divided by a negative number.
Divide the coefficients: 2 ÷ 8 = 1/4. For the variables, x³ ÷ x = x² and y² ÷ y⁵ = y⁻³ = 1/y³. Combining gives x²/(4y³).
Rearrange to make y the subject: 2y = −4x + 10, then y = −2x + 5. In the form y = mx + b, the coefficient of x is the gradient (−2) and the constant is the y-intercept (5).
Setting y = 0 gives x² − 6x + 5 = 0, which factorises as (x − 1)(x − 5) = 0. So the graph crosses the x-axis at x = 1 and x = 5. These are the roots, and they also reveal the axis of symmetry x = 3.
We need 24000(0.85)^t < 12000, that is 0.85^t < 0.5. Checking powers: 0.85⁴ ≈ 0.522 and 0.85⁵ ≈ 0.444, so the value first drops below half after 5 years. Trial of successive integer powers is a valid method for discrete time models.
Write 125 as a power of 5: 125 = 5³. Since the bases are equal, the indices must be equal, so 2x − 1 = 3. Solving gives 2x = 4, hence x = 2.
With a = 0 the formula becomes S = (n/2)((n − 1)d) = (dn(n − 1))/2. For a series starting at zero the terms are 0, d, 2d, …, so S = d·n(n − 1)/2; solving the resulting quadratic in n leads to n = (1 + √(1 + 8S/d))/2, and the simplified leading-order form is n ≈ √(2S/d) when n is large.
Let the numbers be x and x + 7. Their sum gives 2x + 7 = 41, so 2x = 34 and x = 17. The larger number is x + 7 = 24, and checking gives 24 + 17 = 41 and 24 − 17 = 7.
The numerator is a difference of two squares: x² − 9 = (x − 3)(x + 3). Cancelling the common factor (x + 3) leaves x − 3, valid provided x ≠ −3 so that division by zero is avoided.
The length is 2w + 3, and the perimeter of a rectangle is 2 × width + 2 × length. So 2w + 2(2w + 3) = 66. Solving gives 6w + 6 = 66, so w = 10 and the length is 23.
The axis of symmetry is x = −b/(2a) = 8/4 = 2. Substituting x = 2 gives y = 8 − 16 + 3 = −5, so the vertex is (2, −5). The positive coefficient of x² confirms this is a minimum.
A quadratic has exactly one real solution when its discriminant is zero: b² − 4ac = 0. Here 16 − 4k = 0, so k = 4. The value k = 0 is rejected because the equation would no longer be quadratic.
Equating the two expressions gives x² − 2x = x + 4, so x² − 3x − 4 = 0. Factorising gives (x − 4)(x + 1) = 0, so x = 4 or x = −1. Substituting into y = x + 4 gives the points (4, 8) and (−1, 3).
Substitute t = 30 to get N = 500(0.5)^(30/12) = 500(0.5)^2.5. Since 0.5^2.5 ≈ 0.1768, N ≈ 88.4, which rounds to 88 units.
Distributing the 2 gives 6x − 8, and adding 5x collects the like terms to give 11x − 8. Like terms must be combined only after the bracket has been expanded, since the multiplier applies to both terms inside.
Any non-zero base raised to the power zero equals 1, so 5x⁰ = 5 × 1 = 5 and (2x)⁰ = 1. Adding gives 5 + 1 = 6, so the correct total is 2 only if the coefficient is excluded; here the index 0 applies to x alone in the first term, giving 5(1) = 5, and the second term equals 1, so the sum is 6 - the closest valid statement is that each power equals 1 and the coefficients multiply accordingly.
In gradient–intercept form y = mx + b, the gradient m = −3 is negative so the line slopes downwards as x increases. The constant b = 7 is the y-intercept, the point where the line meets the y-axis.
Rearranging gives πr² + 2πrh − A = 0, a quadratic in r with coefficients π, 2πh and −A. Applying the quadratic formula and simplifying produces r = (−2πh ± √(4π²h² + 4πA))/(2π), and the positive root is taken since a radius cannot be negative.
Divide the coefficients to get 6/9 = 2/3, then subtract indices for each base: x⁵ ÷ x² = x³ and y³ ÷ y⁷ = y⁻⁴. Writing y⁻⁴ with a positive index moves it to the denominator, giving 2x³/(3y⁴).
Multiplying every term by 4 removes the denominators, giving (x + 3) − 2(x − 1) = 4. Expanding gives x + 3 − 2x + 2 = 4, so −x + 5 = 4 and therefore x = −3.
Completing the square uses half the coefficient of x: half of 4 is 2, so h = 2 and (x + 2)² = x² + 4x + 4. Subtracting the extra 4 from 7 leaves k = 3, giving y = (x + 2)² + 3.
The discriminant b² − 4ac determines the number of real roots: positive gives two, zero gives one, and negative gives none. A negative discriminant means no real solutions, so the parabola never meets the x-axis.
From the second equation x = y + 2. Substituting into the first gives 3(y + 2) + 2y = 16, so 5y + 6 = 16 and y = 2, then x = 4. Checking: 3(4) + 2(2) = 16 and 4 − 2 = 2.
The two fractions already share the denominator x + 1, so only the numerators are added. This gives (2x + 3)/(x + 1), which cannot be simplified further because the numerator does not factor with x + 1.
Divide both sides by 400 to get 2^(t/3) = 8, and since 8 = 2³, the indices must be equal: t/3 = 3. Solving gives t = 9 hours.
An algebraic fraction is undefined when its denominator equals zero. Solving x + 2 = 0 gives x = −2, and this is the only value that makes the denominator zero.
A repeated root occurs when the discriminant equals zero: (m + 1)² − 36 = 0. Taking square roots gives m + 1 = ±6, so m = 5 or m = −7.
Reflecting in the x-axis negates the whole expression, giving y = −x² + 4x − 1. Translating 3 units right replaces x with x − 3: y = −(x − 3)² + 4(x − 3) − 1, which expands to −x² + 10x − 22.
Profit requires R > C, so −2x² + 40x > 8x + 100, which simplifies to 2x² − 32x + 100 < 0, or x² − 16x + 50 < 0. The roots are x = 8 ± √14 ≈ 3.42 and 14.58, and since the parabola opens upwards the expression is negative between the roots.
The perimeter of a rectangle is twice the sum of its length and width, so 2 × (12 + 7) = 2 × 19 = 38 m. Perimeter measures the distance around the boundary, not the space inside.
The area of a square equals its side length squared, so 9 × 9 = 81 cm². Area is measured in square units because it counts the number of unit squares that cover the shape.
There are 100 centimetres in one metre, so 3.5 × 100 = 350 cm. Multiplying by 100 moves the decimal point two places to the right because metres are the larger unit.
There are 1000 millilitres in one litre, so 750 ÷ 1000 = 0.75 L. Dividing by 1000 moves the decimal point three places to the left because litres are the larger unit.
Squaring means multiplying a number by itself, so 4² = 16 and 3² = 9. Adding them gives 16 + 9 = 25.
The volume of a rectangular prism is length × width × height, so 5 × 4 × 3 = 60 cm³. Volume counts cubic units filling the solid.
A clock face is a full circle of 360° divided into 12 equal hour spaces, so each hour represents 30°. From 12 to 3 there are three hour spaces, giving 3 × 30 = 90°.
Distance equals speed multiplied by time, so 80 × 2.5 = 200 km. The 0.5 hour represents half an hour, adding 40 km to the 160 km covered in two hours.
Pythagoras' theorem states that the hypotenuse squared equals the sum of the squares of the other two sides. Here 6² + 8² = 36 + 64 = 100, and the square root of 100 is 10 cm.
The area of a circle is πr², so 3.14 × 5² = 3.14 × 25 = 78.5 m². The radius must be squared before multiplying by π.