What value(s) of [tex]$x$[/tex] make the equation [tex]$x^2-18 x+81=0$[/tex] true?
A. 0, -9, 9
B. -9, 9
C. 9
D. 0, 9
Answer (1)
Factor the quadratic equation: x 2 − 18 x + 81 = ( x − 9 ) ( x − 9 ) = 0 .
Set each factor equal to zero: x − 9 = 0 .
Solve for x : x = 9 .
The solution to the quadratic equation is 9 .
Explanation
Understanding the Problem We are given the quadratic equation x 2 − 18 x + 81 = 0 and asked to find the value(s) of x that make the equation true. This means we need to solve for x .
Factoring the Quadratic Equation We can solve this quadratic equation by factoring. We are looking for two numbers that multiply to 81 and add up to -18. These numbers are -9 and -9. So, we can factor the quadratic equation as follows: x 2 − 18 x + 81 = ( x − 9 ) ( x − 9 ) = 0
Solving for x Now, we set each factor equal to zero and solve for x :
x − 9 = 0 x = 9
Finding the Solution Since both factors are the same, we have only one solution for x , which is x = 9 .
Examples
Quadratic equations are useful in many real-world scenarios, such as calculating the trajectory of a ball, determining the dimensions of a garden, or modeling the growth of a population. For example, if you want to build a rectangular garden with an area of 81 square feet and you want the length to be 18 feet less than the square of the width, you can use a quadratic equation to find the dimensions of the garden.
