Question

complete the square. find the vertex and roots.
ex: 2(x+4)^2-12=0,(-4,12),-4-2sqrt(3),-4+2sqrt(3)
-x² + 10x - 22 = 0 type your answer...

Explanation:

Step1: Rewrite the equation

First, let's rewrite the given quadratic equation \(-x^{2}+10x - 22=0\). Multiply both sides by \(- 1\) to make the coefficient of \(x^{2}\) positive:
\(x^{2}-10x + 22 = 0\)
Then, move the constant term to the right - hand side:
\(x^{2}-10x=-22\)

Step2: Complete the square

To complete the square for the expression \(x^{2}-10x\), we take half of the coefficient of \(x\), which is \(\frac{- 10}{2}=-5\), and then square it: \((-5)^{2} = 25\).
Add 25 to both sides of the equation \(x^{2}-10x=-22\):
\(x^{2}-10x + 25=-22 + 25\)
The left - hand side can be written as a perfect square: \((x - 5)^{2}=3\)

Step3: Find the vertex

For a quadratic function in the form \(y=a(x - h)^{2}+k\), the vertex is \((h,k)\). Our equation after completing the square is \((x - 5)^{2}-3 = 0\) (or \(y=(x - 5)^{2}-3\) if we consider \(y=-x^{2}+10x - 22\) and rewrite it as \(y=-(x^{2}-10x + 22)=-(x^{2}-10x+25 - 3)=-(x - 5)^{2}+3\)). Wait, we made a mistake in the sign earlier. Let's start over with the original equation \(y=-x^{2}+10x - 22\).
Factor out the coefficient of \(x^{2}\) from the first two terms: \(y=-(x^{2}-10x)-22\)
Complete the square inside the parentheses: \(x^{2}-10x=(x - 5)^{2}-25\)
So \(y=-( (x - 5)^{2}-25)-22=- (x - 5)^{2}+25 - 22=- (x - 5)^{2}+3\)
So the vertex form of the quadratic function \(y = -x^{2}+10x - 22\) is \(y=-(x - 5)^{2}+3\). The vertex of a parabola in the form \(y=a(x - h)^{2}+k\) is \((h,k)\), so the vertex is \((5,3)\)

Step4: Find the roots

We start from the equation \(-x^{2}+10x - 22 = 0\), or \(x^{2}-10x + 22 = 0\) (after multiplying by \(-1\)). Using the completed square form \((x - 5)^{2}=3\) (from the equation \(x^{2}-10x+25 = 3\) which is \(x^{2}-10x+22 = 0\) rewritten).
Take the square root of both sides: \(x - 5=\pm\sqrt{3}\)
Solve for \(x\): \(x = 5\pm\sqrt{3}\)

Answer:

After completing the square, the equation is \(-(x - 5)^{2}+3 = 0\) (or \((x - 5)^{2}=3\) when we multiply by \(- 1\) and rearrange). The vertex is \((5,3)\) and the roots are \(x = 5+\sqrt{3}\) and \(x = 5-\sqrt{3}\)