Sometimes students shy away from completing the square and...

Sometimes students shy away from completing the square and using the square root property to solve a quadratic equation. However, sometimes this process leads to a simple solution. For Exercises $87-88,$ solve the equations two ways. a. Solve the equation by completing the square and applying the square root property. b. Solve the equation by applying the quadratic formula. c. Which technique was easier for you? $$x^{2}-10 x=-27$$

Sometimes students shy away from completing the square and using the square root property to solve a quadratic equation. However, sometimes this process leads to a simple solution. For Exercises $87-88,$ solve the equations two ways.
a. Solve the equation by completing the square and applying the square root property.
b. Solve the equation by applying the quadratic formula.
c. Which technique was easier for you?
$$x^{2}-10 x=-27$$

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Key Concepts

Completing the Square

Completing the square is a method used to solve quadratic equations by rewriting the equation in the form (x - h)² = k. This involves manipulating the equation to form a perfect square trinomial on one side and then solving for x by applying the square root property. It is especially useful for understanding the derivation of the quadratic formula and for problems where the quadratic does not factor easily.

Square Root Property

The square root property is a technique that is applied once an equation has been rewritten in the form (expression)² = number. By taking the square root of both sides of the equation, one obtains two potential solutions, corresponding to the positive and negative square roots. This property is fundamental in solving quadratics after completing the square.

Quadratic Formula

The quadratic formula provides a direct method for solving any quadratic equation expressed in the standard form ax² + bx + c = 0. By substituting the coefficients into the formula x = (-b ± ?(b² - 4ac))/(2a), one can determine the roots of the quadratic. It serves as a universal tool, particularly when factoring is difficult or when the equation does not readily lend itself to other methods like completing the square.

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