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In an arithmetic sequence the nth term is found from a_n = a_1 + (n - 1)d. Substituting a_1 = 7, d = 4 and n = 15 gives 7 + 14*4, which equals 63.
Square each factor inside the bracket: (2x^3)^2 = 4x^6. Multiplying by 3x^2 then gives coefficient 4*3 = 12 and index 6 + 2 = 8, so the result is 12x^8.
The midpoint is found by averaging the x-coordinates and the y-coordinates separately. This gives ((-3 + 7)/2, (5 + -1)/2) = (2, 2).
Subtract 5 from both sides to get -2x < 6. Dividing by -2 reverses the inequality sign, so x > -3.
From x - y = 2, x = y + 2. Substituting into 3x + 2y = 16 gives 3(y + 2) + 2y = 16, so 5y + 6 = 16 and y = 2. Then x = 2 + 2 = 4.
The expression factorises as (x - 3)(x - 4) = 0 because -3 and -4 multiply to 12 and add to -7. Setting each factor equal to zero gives x = 3 or x = 4.
The first term is 5 and the common difference is 4. Using S_n = n/2[2a + (n - 1)d] gives S_20 = 10[10 + 19*4] = 10*86 = 860.
Subtract u from both sides to obtain v - u = at^2. Divide by a and then take the positive square root, giving t = sqrt((v - u)/a).
For two distinct real roots the discriminant b^2 - 4ac must be greater than zero. Here b^2 - 4ac = k^2 - 4*2*8 = k^2 - 64, so k^2 > 64, which gives k < -8 or k > 8.
Let the first term be a and the common ratio be r. From ar = 6 and ar^4 = 48, dividing gives r^3 = 8, so r = 2 and a = 3. The sum of six terms is 3(2^6 - 1)/(2 - 1) = 3*63 = 189.
The fractional index 2/3 means take the cube root of 8 and then square it: 8^(1/3) = 2, and 2^2 = 4. Equivalently, 8^(2/3) = (∛8)^2 = 4.
This is a difference of two squares, since 4x^2 = (2x)^2 and 25 = 5^2. The pattern a^2 - b^2 = (a - b)(a + b) gives (2x - 5)(2x + 5).
Rearrange the equation into the form y = mx + c: 4y = -3x + 12, so y = -3/4 x + 3. The gradient is the coefficient of x, which is -3/4.
Factorise the numerator as (x - 2)(x + 2) and the denominator as (x + 3)(x - 2). Cancel the common factor (x - 2), leaving (x + 2)/(x + 3), with x ≠ 2 and x ≠ -3.
Write both sides with base 3: 9^x = (3^2)^x = 3^(2x), and 27^(x - 1) = (3^3)^(x - 1) = 3^(3x - 3). Equating exponents gives 2x = 3x - 3, so x = 3.
Factorise x^2 - 4x - 5 as (x - 5)(x + 1). The parabola opens upwards and is below the x-axis between its roots -1 and 5, so the inequality holds for -1 < x < 5.
Combine the logs: log_2[x(x - 2)] = 3, so x(x - 2) = 2^3 = 8. This gives x^2 - 2x - 8 = 0, or (x - 4)(x + 2) = 0. Since logs require x > 2, the only valid solution is x = 4.
Complete the square: x^2 - 6x + 11 = (x - 3)^2 + 2. Since (x - 3)^2 is never negative, the smallest value occurs when it is zero, giving a minimum of 2.
Multiply numerator and denominator by the conjugate 2 + √5. The denominator becomes 4 - 5 = -1, while the numerator becomes 6 + 3√5 + 2√5 + 5 = 11 + 5√5, so the result is -(11 + 5√5).
Set the expressions equal: x + 1 = x^2 - 1, which rearranges to x^2 - x - 2 = 0. Factorising gives (x - 2)(x + 1) = 0, so the x-coordinates are 2 and -1. Their sum is 1.
Square each factor in the numerator to get 16a^6 b^4, then divide by 8a^2 b^5. Subtract the exponents: a^(6-2)=a^4 and b^(4-5)=b^(-1). This leaves 2a^4/b.
Standard form is A × 10^n with 1 ≤ A < 10. Moving the decimal point 6 places to the right gives 7.5, so the exponent is -6. Thus 0.0000075 = 7.5 × 10^-6.
The highest common factor of 10x^2 and 15x is 5x. Taking 5x out gives 5x(2x - 3). The expression inside the brackets has no further common factor.
Squaring both sides gives h^2 = a^2 + b^2. Rearranging gives b^2 = h^2 - a^2, and since b is positive, b = √(h^2 - a^2).
Substitute y = 2x + 1 into x^2 + y^2 = 13 to get 5x^2 + 4x - 12 = 0. This gives x = -2 or x = 6/5, and substituting back gives y = -3 or y = 17/5. The solution pairs are therefore (-2, -3) and (6/5, 17/5).
Multiply every term by 12, the common denominator, to obtain 4(2x - 1) - 3(x + 2) = 12. Expanding gives 8x - 4 - 3x - 6 = 12, so 5x = 22 and x = 22/5.
Using the quotient law, log_3 18 - log_3 2 = log_3 9 = 2. Then log_3 9 = 2 as well, so the total is 2 + 2 = 4.
For real roots the discriminant must be non-negative: (k - 2)^2 - 16 ≥ 0. This factorises as (k - 6)(k + 2) ≥ 0, whose solution is k ≤ -2 or k ≥ 6.
Bring all terms to one side: (x + 1)/(x - 3) - 2 = (7 - x)/(x - 3) ≥ 0. The critical values are x = 3, where the expression is undefined, and x = 7, where it equals zero. Testing intervals gives 3 < x ≤ 7.
By the remainder theorem, P(1) = 0 gives 2 + a + b - 6 = 0, so a + b = 4. Also P(-1) = -12 gives -2 + a - b - 6 = -12, so a - b = -4. Solving these equations gives a = 0 and b = 4.
The sequence increases by 4 each time, so the common difference is 4. Using a_n = a + (n-1)d gives a_20 = 7 + 19 × 4 = 83.
The gradient is (10 - 4)/(3 - 1) = 3. Using y - 4 = 3(x - 1) simplifies to y = 3x + 1.
Write each surd with the largest square factor removed: √50 = 5√2 and √18 = 3√2. Adding like surds gives 5√2 + 3√2 = 8√2.
Apply the power to each factor: 2⁻² = 1/4, (a⁻³)⁻² = a⁶, and (b²)⁻² = b⁻⁴. So the expression is a⁶/(4b⁴).
Use the power law: 3 log₂ a = log₂(a³). Then subtract logs to divide: log₂(a³) - log₂ b = log₂(a³/b).
Use the common denominator (x + 2)(x - 1). The numerator is 3(x - 1) + 2(x + 2) = 3x - 3 + 2x + 4 = 5x + 1.
Let y = 3x - 7 and swap x and y to get x = 3y - 7. Solving for y gives y = (x + 7)/3.
Add equations 1 and 3 to eliminate z: 2x + 3y = 8. Subtract equation 1 from equation 2 to get x - 2y = -3. Solving gives x = 1, y = 2, and then z = 6 - 1 - 2 = 3.
Dividing the fifth term by the second gives r³ = (3/4)/6 = 1/8, so r = 1/2. Then a = 6/(1/2) = 12 and S∞ = a/(1 - r) = 12/(1/2) = 24.
Multiply both sides by cx + d: y(cx + d) = ax + b. Expand and collect x terms: ycx - ax = b - yd, so x(yc - a) = b - yd and x = (b - yd)/(yc - a).
Expand by multiplying each term in the first bracket by each term in the second: 2x(x + 5) - 3(x + 5) = 2x^2 + 10x - 3x - 15. Collecting the like terms 10x and -3x gives 2x^2 + 7x - 15.
Cubing gives (3x^2)^3 = 27x^6, because both the coefficient 3 and the index 2 are raised to the power 3. Dividing by 9x^4 gives (27/9) times x^(6-4), which simplifies to 3x^2.
Write both sides with base 2: 4^(x + 1) = 2^(2x + 2) and 8^(x - 1) = 2^(3x - 3). Equating the exponents gives 2x + 2 = 3x - 3, so x = 5, and checking shows 4^6 = 8^4 = 4096.
Multiply numerator and denominator by the conjugate sqrt(5) + 1, giving 6(sqrt(5) + 1) / (5 - 1) = 6(sqrt(5) + 1)/4. Cancelling the common factor 2 gives 3(sqrt(5) + 1)/2, which has a rational denominator.
Combining the logarithms gives log_2[x(x - 2)] = 3, so x^2 - 2x = 8 and hence x^2 - 2x - 8 = 0. Factorising gives (x - 4)(x + 2) = 0, and the domain condition x > 2 leaves only x = 4.
Use the sum formula S_n = n/2[2a + (n - 1)d] with a = 5, d = 3 and n = 20. This gives S_20 = 10[10 + 19 times 3] = 10 times 67 = 670.
The nth term is 3 times 2^(n - 1), so the condition is 2^(n - 1) > 1000/3, which is about 333.3. Since 2^8 = 256 and 2^9 = 512, the smallest possible index is n - 1 = 9, giving n = 10.
Factorising gives (x + 4)(x - 2) > 0. The parabola y = x^2 + 2x - 8 opens upwards with roots at x = -4 and x = 2, so the expression is positive outside the roots, that is x < -4 or x > 2.
Substitute u = 2^x so the equation becomes u^2 - 5u + 4 = 0, which factorises as (u - 1)(u - 4) = 0. Thus u = 1 or u = 4, and solving 2^x = 1 and 2^x = 4 gives x = 0 or x = 2.
Use the fractional-index rule a^(m/n) = (nth root of a)^m. So 27^(2/3) = (cube root of 27)^2 = 3^2 = 9, and 16^(1/2) = 4. Adding gives 9 + 4 = 13.