What is the result of applying the square root property of equality to the equation (x + b/2a)² = -4ac + b²/4a²?

Understand the Problem

The question is asking what result is obtained when applying the square root property of equality to a specific equation derived during the process of completing the square for the quadratic formula. The user needs to evaluate the context of the mathematical equation provided.

Answer

$$ x = \frac{-b \pm \sqrt{-4ac + b^2}}{2a} $$

Steps to Solve

Identify the equation's form

The equation given is $ \left( x + \frac{b}{2a} \right)^2 = \frac{-4ac + b^2}{4a^2} $. This indicates that we have a perfect square on the left-hand side.
Apply the square root property of equality

Using the square root property, we take the square root of both sides: $$ x + \frac{b}{2a} = \pm \sqrt{\frac{-4ac + b^2}{4a^2}} $$
Simplify the equation

Now simplify the right side. The square root can be separated: $$ x + \frac{b}{2a} = \pm \frac{\sqrt{-4ac + b^2}}{2a} $$
Isolate x

To isolate $x$, subtract $\frac{b}{2a}$ from both sides: $$ x = -\frac{b}{2a} \pm \frac{\sqrt{-4ac + b^2}}{2a} $$
Combine fractions

Combine the terms on the right side: $$ x = \frac{-b \pm \sqrt{-4ac + b^2}}{2a} $$

The result of applying the square root property of equality to the given equation is: $$ x = \frac{-b \pm \sqrt{-4ac + b^2}}{2a} $$

More Information

This expression represents the quadratic formula, which is used to find the roots of a quadratic equation in standard form, $ax^2 + bx + c = 0$. The solutions are determined using the coefficients $a$, $b$, and $c$ from the quadratic equation.

Tips

Forget to apply the $\pm$ sign: When taking the square root of both sides, it's crucial to remember that this results in two possible solutions (positive and negative).
Mismanage algebraic signs: Pay careful attention to signs when isolating ( x ).

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