Solve $x^2=8+x$

Select the correct choice below and, if necessary, fill in the answer box to complete your answer.
A. The solution set is $\square$
(Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as ne
B. The equation has no real solution:

Rewrite the equation in standard quadratic form: x 2 − x − 8 = 0 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c ​ ​ .
Substitute a = 1 , b = − 1 , and c = − 8 into the formula and simplify.
The solution set is 2 1 − 33 ​ ​ , 2 1 + 33 ​ ​ ​ .

Explanation

Understanding the Problem We are given the quadratic equation x 2 = 8 + x . Our goal is to find the solution set for x . To do this, we will rewrite the equation in the standard quadratic form and then use the quadratic formula to find the solutions.

Rewriting the Equation First, rewrite the equation in the standard quadratic form a x 2 + b x + c = 0 . Subtract x and 8 from both sides of the equation to get: x 2 − x − 8 = 0 Here, a = 1 , b = − 1 , and c = − 8 .

Applying the Quadratic Formula Now, apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c ​ ​ Substitute the values of a , b , and c into the formula: x = 2 ( 1 ) − ( − 1 ) ± ( − 1 ) 2 − 4 ( 1 ) ( − 8 ) ​ ​ Simplify the expression: x = 2 1 ± 1 + 32 ​ ​ x = 2 1 ± 33 ​ ​ So, the two solutions are: x 1 ​ = 2 1 − 33 ​ ​ ≈ − 2.372 x 2 ​ = 2 1 + 33 ​ ​ ≈ 3.372

Stating the Solution Set The solution set is the set of these two solutions. Therefore, the solution set is: { 2 1 − 33 ​ ​ , 2 1 + 33 ​ ​ }


Examples
Quadratic equations are used in various real-life situations, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its area and a relationship between its sides, or modeling the growth of a population. For example, if you want to build a rectangular garden with an area of 100 square meters and the length must be 5 meters more than the width, you can use a quadratic equation to find the dimensions of the garden.

Answered by GinnyAnswer | 2025-07-08

The solutions to the equation x 2 = 8 + x are 2 1 − 33 ​ ​ and 2 1 + 33 ​ ​ . Therefore, the chosen option is A. The solution set is { 2 1 − 33 ​ ​ , 2 1 + 33 ​ ​ } .
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Answered by Anonymous | 2025-07-14