Sure! Let's solve the quadratic equation [tex]\( y = x^2 + 10x + 10 \)[/tex] by completing the square. Here are the steps:
1. Write down the original quadratic equation:
[tex]\[
y = x^2 + 10x + 10
\][/tex]
2. Isolate the quadratic and linear terms:
[tex]\[
y - 10 = x^2 + 10x
\][/tex]
3. Complete the square:
- To complete the square, we need to add and subtract a constant inside the equation such that it forms a perfect square trinomial on the right-hand side.
- Take the coefficient of [tex]\( x \)[/tex], which is 10, divide by 2, and then square it:
[tex]\[
\left(\frac{10}{2}\right)^2 = 5^2 = 25
\][/tex]
4. Add and subtract this constant inside the equation:
[tex]\[
y - 10 = x^2 + 10x + 25 - 25
\][/tex]
5. Rewrite the right-hand side as a perfect square trinomial:
[tex]\[
y - 10 = (x + 5)^2 - 25
\][/tex]
6. Simplify the equation by combining constants on the left-hand side:
[tex]\[
y = (x + 5)^2 - 15
\][/tex]
So, the quadratic expression [tex]\( y = x^2 + 10x + 10 \)[/tex], when written in its completed square form, is:
[tex]\[
y = (x + 5)^2 - 15
\][/tex]
This expression shows the quadratic in a way that highlights its vertex form, making it easier to analyze its properties, such as the vertex, which is at [tex]\( (-5, -15) \)[/tex].
