Which are the solutions of the quadratic equation? [tex]x^2=7 x+4[/tex]

A. [tex]\frac{-7-\sqrt{65}}{2}, \frac{-7+\sqrt{65}}{2}[/tex]
B. [tex]{-7,0}[/tex]
C. [tex]\frac{7-\sqrt{65}}{2}, \frac{7+\sqrt{65}}{2}[/tex]
D. [tex]{7,0}[/tex]

Rewrite the given quadratic equation in the standard form: x 2 − 7 x − 4 = 0 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c ​ ​ , where a = 1 , b = − 7 , and c = − 4 .
Substitute the values into the formula and simplify: x = 2 7 ± 49 + 16 ​ ​ = 2 7 ± 65 ​ ​ .
The solutions are: 2 7 − 65 ​ ​ , 2 7 + 65 ​ ​ ​ .

Explanation

Understanding the Problem We are given the quadratic equation x 2 = 7 x + 4 . Our goal is to find the solutions to this equation from the given options.

Rewriting the Equation First, we need to rewrite the equation in the standard quadratic form, which is a x 2 + b x + c = 0 . Subtracting 7 x and 4 from both sides of the equation, we get: x 2 − 7 x − 4 = 0

Identifying Coefficients and the Quadratic Formula Now we can identify the coefficients: a = 1 , b = − 7 , and c = − 4 . We will use the quadratic formula to find the solutions for x . The quadratic formula is: x = 2 a − b ± b 2 − 4 a c ​ ​

Substituting Values Substitute the values of a , b , and c into the quadratic formula: x = 2 ( 1 ) − ( − 7 ) ± ( − 7 ) 2 − 4 ( 1 ) ( − 4 ) ​ ​

Simplifying the Expression Simplify the expression: x = 2 7 ± 49 + 16 ​ ​ x = 2 7 ± 65 ​ ​

Finding the Solutions So the two solutions are: x 1 ​ = 2 7 − 65 ​ ​ x 2 ​ = 2 7 + 65 ​ ​ Comparing these solutions with the given options, we see that the correct answer is 2 7 − 65 ​ ​ , 2 7 + 65 ​ ​ .


Examples
Quadratic equations are incredibly useful in various real-world scenarios. For instance, they can model the trajectory of a ball thrown in the air, helping to determine its maximum height and range. They are also used in engineering to design bridges and arches, ensuring structural stability. In finance, quadratic equations can help model investment growth and calculate optimal portfolio allocations. Understanding how to solve quadratic equations provides valuable tools for analyzing and predicting outcomes in these diverse fields.

Answered by GinnyAnswer | 2025-07-03

To solve the equation x 2 = 7 x + 4 , we rewrite it in standard form x 2 − 7 x − 4 = 0 and apply the quadratic formula. The solutions obtained are 2 7 − 65 ​ ​ and 2 7 + 65 ​ ​ . The correct answer is option C.
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Answered by Anonymous | 2025-07-04