Question Find the value of $y$ in the equation $y = 2x^2 + 12x + 5$ by completing the square and solving for $y$ .

Studdy Solution

STEP 1

Assumptions
1. We are given a quadratic equation in the form $y = ax^2 + bx + c$ .
2. We want to rewrite the equation in the form of a perfect square, $y = a(x-h)^2 + k$ , where $(x-h)^2$ is the perfect square trinomial.
3. To create a perfect square trinomial, we will need to complete the square, which involves adding and subtracting the square of half the coefficient of $x$ within the parentheses.
4. After completing the square, we will simplify the equation to solve for $y$ .

STEP 2

Identify the coefficients of the quadratic equation.
Given the quadratic equation $y = 2x^2 + 12x + 5$ , we have:
$a = 2$ , $b = 12$ , $c = 5$ .

STEP 3

Divide the coefficient $b$ by 2 and square the result to find the number to add and subtract to complete the square.
$\left(\frac{b}{2}\right)^2 = \left(\frac{12}{2}\right)^2 = 6^2 = 36$

STEP 4

Add and subtract the number found in STEP_3 inside the equation to create a perfect square trinomial. We will add and subtract $36a$ to ensure the equation remains balanced since the coefficient of $x^2$ is $2$ and not $1$ .
$y = 2x^2 + 12x + 2 \cdot 36 - 2 \cdot 36 + 5$

STEP 5

Simplify the equation by combining like terms.
$y = 2x^2 + 12x + 72 - 72 + 5$
$y = 2x^2 + 12x + 72 + (5 - 72)$
$y = 2x^2 + 12x + 72 - 67$

STEP 6

Factor the perfect square trinomial.
$y = 2(x^2 + 6x + 36) - 67$

STEP 7

Recognize that the trinomial $x^2 + 6x + 36$ is a perfect square.
$y = 2(x + 3)^2 - 67$

STEP 8

Now the equation is in the form of a perfect square plus a constant, and we have completed the square.
The simplified equation for $y$ is:
$y = 2(x + 3)^2 - 67$
This is the equation with a perfect square trinomial and the value for $y$ after simplifying.

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