Which are the solutions of $x^2=-5 x+8 ?$

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\frac{-5-\sqrt{57}}{2}, \frac{-5+\sqrt{57}}{2}$

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\frac{-5-\sqrt{7}}{2}, \frac{-5+\sqrt{7}}{2}$

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\frac{5-\sqrt{57}}{2}, \frac{5+\sqrt{57}}{2}$

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\frac{5-\sqrt{7}}{2}, \frac{5+\sqrt{7}}{2}$

Rewrite the equation in standard form: x 2 + 5 x − 8 = 0 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c ​ ​ .
Substitute a = 1 , b = 5 , c = − 8 into the formula: x = 2 ( 1 ) − 5 ± 5 2 − 4 ( 1 ) ( − 8 ) ​ ​ .
Simplify to find the solutions: x = 2 − 5 ± 57 ​ ​ , so the solutions are 2 − 5 − 57 ​ ​ , 2 − 5 + 57 ​ ​ ​ .

Explanation

Problem Analysis We are given the quadratic equation x 2 = − 5 x + 8 . Our goal is to find the solutions for x . To do this, we will rewrite the equation in the standard quadratic form and then apply the quadratic formula.

Rewriting the Equation First, we rewrite the equation in the standard form a x 2 + b x + c = 0 . Adding 5 x to both sides and subtracting 8 from both sides, we get: x 2 + 5 x − 8 = 0 Now we can identify the coefficients: a = 1 , b = 5 , and c = − 8 .

Applying the Quadratic Formula Next, we apply the quadratic formula, which states that for an equation of the form a x 2 + b x + c = 0 , the solutions are given by: x = 2 a − b ± b 2 − 4 a c ​ ​ Substituting the values of a , b , and c into the quadratic formula, we get: x = 2 ( 1 ) − 5 ± 5 2 − 4 ( 1 ) ( − 8 ) ​ ​

Simplifying the Expression Now, we simplify the expression under the square root: 5 2 − 4 ( 1 ) ( − 8 ) = 25 + 32 = 57 So, the solutions become: x = 2 − 5 ± 57 ​ ​

Final Solutions Therefore, the two solutions are: x 1 ​ = 2 − 5 − 57 ​ ​ and x 2 ​ = 2 − 5 + 57 ​ ​ These are the roots of the given quadratic equation.


Examples
Quadratic equations are used in various real-life scenarios, 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 or decay of populations. 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 | 2025-07-03

The solutions to the equation x 2 = − 5 x + 8 are 2 − 5 − 57 ​ ​ and 2 − 5 + 57 ​ ​ . Therefore, the correct options are: 2 − 5 − 57 ​ ​ , 2 − 5 + 57 ​ ​ .
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Answered by Anonymous | 2025-07-04