Write the following expression in vertex form by completing the square. 𝑦 = 𝑥2 − 12𝑥 + 7
To complete the square, we need to take half of the coefficient of the x-term and square it. Then, we can add and subtract this value inside the parentheses containing the x-terms in the quadratic.
First, let's factor out any common factor from the x-terms:
𝑦 = 𝑥^2 − 12𝑥 + 7
𝑦 = 1(𝑥^2 − 12𝑥) + 7
Now, we can see that the coefficient of the x-term is -12, so we need to add and subtract (12/2)^2 = 36 inside the parentheses:
𝑦 = 1(𝑥^2 − 12𝑥 + 36 - 36) + 7
𝑦 = 1(𝑥 - 6)^2 - 29
So the vertex form of the equation is y = (𝑥 - 6)^2 - 29.
To write the given expression 𝑦 = 𝑥^2 − 12𝑥 + 7 in vertex form by completing the square, follow these steps:
Step 1: Group the x-terms together:
𝑦 = (𝑥^2 − 12𝑥) + 7
Step 2: Complete the square by adding and subtracting the square of half the coefficient of the x-term. In this case, the coefficient is -12, so half of that is -6. The square of -6 is 36. Add and subtract 36 inside the parentheses:
𝑦 = (𝑥^2 − 12𝑥 + 36 - 36) + 7
Step 3: Rearrange the terms inside the parentheses:
𝑦 = (𝑥^2 − 12𝑥 + 36) - 36 + 7
Step 4: Simplify inside the parentheses:
𝑦 = (𝑥 - 6)^2 - 29
The vertex form of the given quadratic expression is 𝑦 = (𝑥 - 6)^2 - 29.