Solve $x^2-8 x=20$ by completing the square. Which is the solution set of the equation?
A. $\{-22,14\}$
B. $\{-14,22\}$
C. $\{-10,2\}$
D. $\{-2,10\}$
Answer (1)
Rewrite the equation x 2 − 8 x = 20 as x 2 − 8 x − 20 = 0 .
Complete the square by adding and subtracting ( 2 − 8 ) 2 = 16 to get ( x − 4 ) 2 − 36 = 0 .
Rewrite the equation as ( x − 4 ) 2 = 36 .
Solve for x by taking the square root: x − 4 = ± 6 , which gives x = 10 or x = − 2 . The solution set is { − 2 , 10 } .
Explanation
Understanding the Problem We are given the quadratic equation x 2 − 8 x = 20 and asked to solve it by completing the square. Our goal is to rewrite the equation in the form ( x − h ) 2 = k , where h and k are constants. This will allow us to easily solve for x by taking the square root of both sides.
Rewriting the Equation First, we rewrite the equation by moving the constant term to the left side: x 2 − 8 x − 20 = 0 .
Completing the Square To complete the square, we need to add and subtract a value to the left side of the equation such that the first three terms form a perfect square. We take half of the coefficient of the x term, which is − 8/2 = − 4 , and square it: ( − 4 ) 2 = 16 . So, we add and subtract 16 to the left side of the equation: x 2 − 8 x + 16 − 16 − 20 = 0 .
Rewriting as a Squared Term Now, we rewrite the first three terms as a squared term: ( x − 4 ) 2 − 16 − 20 = 0 .
Simplifying the Equation Combine the constant terms: ( x − 4 ) 2 − 36 = 0 .
Isolating the Squared Term Add 36 to both sides of the equation: ( x − 4 ) 2 = 36 .
Taking the Square Root Take the square root of both sides: x − 4 = ± 6 .
Solving for x Solve for x : x = 4 ± 6 . This gives us two possible solutions: x 1 = 4 + 6 = 10 and x 2 = 4 − 6 = − 2 .
Final Answer Therefore, the solution set of the equation is { − 2 , 10 } .
Examples
Completing the square is a useful technique in various real-world scenarios. For example, consider optimizing the area of a rectangular garden given a fixed perimeter. By expressing the area in terms of one side and completing the square, you can find the dimensions that maximize the garden's area. This method is also used in physics to analyze projectile motion, where completing the square helps determine the maximum height reached by a projectile.
