Question

(a) Express the quadratic function f(x) = 2x² + 32x + 126 in standard form.

a) f(x) = 2x² + 32x + 126
b) f(x) = 2(x + 8)² + 62
c) f(x) = 2(x + 4)² + 110
d) f(x) = 2(x + 16)² + 94

Answer 1

The quadratic function f(x) = 2x² + 32x + 126 in standard form is f(x) = 2(x + 16)² - 386, obtained by completing the square. The correct form does not match any of the given options, indicating a possible typo.

Step-by-step explanation:

To express the quadratic function f(x) = 2x² + 32x + 126 in standard form, we need to complete the square. Completing the square involves creating a perfect square trinomial from the quadratic and linear terms so that the equation can be expressed in the form of a(x - h)² + k, where (x - h)² is the perfect square, a is the coefficient of the x² term, and k is the constant. Here's the step-by-step process:

1. Divide the coefficient of the x-term by 2, which is 32/2 = 16.
2. Square this value to get 16² = 256.
3. Add and subtract this square value (256) inside the quadratic equation to balance it. However, because the original quadratic equation is multiplied by 2, we need to account for this factor when adding and subtracting, which means we add and subtract 2 × 256 = 512.
4. Now rewrite the quadratic equation as f(x) = 2x² + 32x + 512 - 512 + 126.
5. Regroup to form a perfect square: f(x) = 2(x² + 16x + 256) - 512 + 126.
6. We recognize (x + 16)² is the perfect square, so we rewrite it as f(x) = 2(x + 16)² + 126 - 512.
7. Finally, subtract 512 from 126 to get -386. The equation in standard form is f(x) = 2(x + 16)² - 386.

Note that the correct answer is not one of the options given in the question, which suggests there might be a typo in the question or answer choices.