Question

In the derivation of the quadratic formula by completing the square, the equation (x+b over 2a)²=-4ac+b² over 4a² is created by forming a perfect square trinomial. What is the result of applying the square root property of equality to this equation?

Answer 1

By applying the square root property to the equation created from completing the square, (x + b/2a)² = (-4ac + b²)/4a², we derive the quadratic formula, x = (-b ± √(b² - 4ac))/(2a).

Step-by-step explanation:

In the derivation of the quadratic formula by completing the square, we manipulate the standard form of the quadratic equation ax² + bx + c = 0 and create a perfect square trinomial.

This results in the equation (x + \frac{b}{2a})² = \frac{-4ac + b²}{4a²}. By applying the square root property of equality to this equation, we take the square root of both sides, which yields x + \frac{b}{2a} = ±\sqrt{\frac{-4ac + b²}{4a²}}, and then simplifying further to isolate x gives us the quadratic formula, x = \frac{-b ± \sqrt{b² - 4ac}}{2a}.