Question

solve the equation for all values of x by completing the square. $x^{2}-16x=-61$

Explanation:

Step1: Find the value to add to complete the square

For the quadratic equation of the form $x^{2}+bx = c$, we add $(\frac{b}{2})^{2}$ to both sides. Here $b=- 16$, so $(\frac{-16}{2})^{2}=64$.
$x^{2}-16x + 64=-61 + 64$

Step2: Rewrite the left - hand side as a perfect square

The left - hand side $x^{2}-16x + 64=(x - 8)^{2}$, and the right - hand side is $3$. So we have $(x - 8)^{2}=3$.

Step3: Solve for x

Take the square root of both sides: $x-8=\pm\sqrt{3}$. Then $x = 8\pm\sqrt{3}$.

Answer:

$x = 8+\sqrt{3},x = 8-\sqrt{3}$