O-Level Quadratic functions: worked solution

Question

The curve y = kx² + 2kx − 3 can be put in the form k(x + b)² − 1. Determine k and b.

The target form k(x + b)² - 1 is a completed square, so complete the square on the given expression. y = kx² + 2kx - 3 Take the factor k out of the two terms that contain x (leave the -3 alone): = k(x² + 2x) - 3 Complete the square inside the bracket. Halve the coefficient of x, which is 2, giving 1, then write x² + 2x = (x + 1)² - 1 (the -1 corrects for the extra 1² that squaring the bracket introduces). Substitute this back in: y = k[(x + 1)² - 1] - 3 Multiply the k through the square bracket: = k(x + 1)² - k - 3 Now compare with the required form k(x + b)² - 1. Compare inside the brackets: (x + b)² must match (x + 1)², so b = 1 Compare the constant terms outside: -k - 3 must equal -1, so -k - 3 = -1 -k = -1 + 3 -k = 2 k = -2 Check by expanding the answer: -2(x + 1)² - 1 = -2(x² + 2x + 1) - 1 = -2x² - 4x - 2 - 1 = -2x² - 4x - 3, which is kx² + 2kx - 3 with k = -2. Correct.