Solve $x^2-24 x=-80$ by completing the square. What is the solution set of the equation?
{2,40}
{4,20}
{5,16}
{8,10}

Rewrite the equation in the form x 2 − 24 x + 80 = 0 .
Complete the square: ( x − 12 ) 2 − 144 + 80 = 0 , which simplifies to ( x − 12 ) 2 = 64 .
Take the square root of both sides: x − 12 = ± 8 .
Solve for x to find the solutions: x = 4 and x = 20 . The solution set is { 4 , 20 } ​ .

Explanation

Problem Analysis We are given the quadratic equation x 2 − 24 x = − 80 . Our goal is to solve this equation by completing the square and identify the correct solution set from the given options.

Rewrite the Equation First, we rewrite the equation by adding 80 to both sides to get: x 2 − 24 x + 80 = 0

Completing the Square To complete the square, we take half of the coefficient of the x term, which is 2 − 24 ​ = − 12 . Then, we square it: ( − 12 ) 2 = 144 .

Add and Subtract Now, we add and subtract 144 to the left side of the equation: x 2 − 24 x + 144 − 144 + 80 = 0

Rewrite as a Square We rewrite the first three terms as a square: ( x − 12 ) 2 − 144 + 80 = 0

Simplify Simplify the equation: ( x − 12 ) 2 − 64 = 0

Isolate Squared Term Isolate the squared term by adding 64 to both sides: ( x − 12 ) 2 = 64

Take Square Root Take the square root of both sides: x − 12 = ± 8

Solve for x Solve for x : x = 12 ± 8 This gives two solutions: x = 12 + 8 = 20 and x = 12 − 8 = 4

Solution Set Therefore, the solution set is { 4 , 20 } .


Examples
Completing the square is a useful technique in various real-world applications. For example, consider optimizing the area of a rectangular garden given a fixed perimeter. By expressing the area in terms of one variable and completing the square, you can find the dimensions that maximize the garden's area. This method is also fundamental in physics for analyzing projectile motion and determining maximum heights or ranges.

Answered by GinnyAnswer | 2025-07-08

The solution to the equation is found by completing the square, resulting in the values of x being 4 and 20 . Thus, the solution set is {4, 20}. Therefore, the correct option is {4, 20}.
;

Answered by Anonymous | 2025-07-12