Question

to complete the square means to add a number which makes an expression into the square of a binomial. here is how we can solve a quadratic equation by completing the square. add 9 to complete the square. x² + 6x + 9 = 7 + 9; x² + 6x + 9 = 16; (x + 3)² = 16; x + 3 = 4 or x + 3 = -4; x = 1 or x = -7. here are some for you to try. x² + 2x = 24; x² - 6x = -5; x² - 10x = 11; x² + 8x = -15

Explanation:

For \(x^{2}+2x = 24\):

Step1: Complete the square

Add \(1\) to both sides (\((\frac{2}{2})^2=1\)). So \(x^{2}+2x + 1=24 + 1\), which is \((x + 1)^{2}=25\).

Step2: Solve for \(x\)

\(x+1 = 5\) or \(x + 1=-5\), so \(x = 4\) or \(x=-6\).

For \(x^{2}-6x=-5\):

Step1: Complete the square

Add \(9\) to both sides (\((\frac{-6}{2})^2 = 9\)). So \(x^{2}-6x+9=-5 + 9\), which is \((x - 3)^{2}=4\).

Step2: Solve for \(x\)

\(x-3 = 2\) or \(x - 3=-2\), so \(x = 5\) or \(x = 1\).

For \(x^{2}-10x = 11\):

Step1: Complete the square

Add \(25\) to both sides (\((\frac{-10}{2})^2=25\)). So \(x^{2}-10x + 25=11+25\), which is \((x - 5)^{2}=36\).

Step2: Solve for \(x\)

\(x-5 = 6\) or \(x - 5=-6\), so \(x=11\) or \(x=-1\).

For \(x^{2}+8x=-15\):

Step1: Complete the square

Add \(16\) to both sides (\((\frac{8}{2})^2 = 16\)). So \(x^{2}+8x+16=-15 + 16\), which is \((x + 4)^{2}=1\).

Step2: Solve for \(x\)

\(x+4 = 1\) or \(x + 4=-1\), so \(x=-3\) or \(x=-5\).

Answer:

For \(x^{2}+2x = 24\):
\(x = 4\) or \(x=-6\)
For \(x^{2}-6x=-5\):
\(x = 5\) or \(x = 1\)
For \(x^{2}-10x = 11\):
\(x=11\) or \(x=-1\)
For \(x^{2}+8x=-15\):
\(x=-3\) or \(x=-5\)