use the quadratic equation x^2-22x=75 to answer the following questions

Question

Part A - suppose an equivalent quadratic equation is written x^2 - 22x + c = 75 + c. What value of c would make this equation a perfect square

Answer 1

To make the equation x^2 - 22x + c = 75 + c a perfect square, we need to complete the square.

The coefficient of x^2 is already 1, so we can focus on completing the square for the x terms.

The formula for completing the square for x^2 - bx is adding (b/2)^2 to both sides. In this case, b = -22, so we will add (-22/2)^2 = (-11)^2 = 121 to both sides:

x^2 - 22x + 121 + c = 75 + c + 121

Simplifying the equation gives us:

x^2 - 22x + 121 + c = 196 + c

The left side of the equation is now a perfect square:

(x - 11)^2 + c = 196 + c

Therefore, c = 196.