Using the quadratic formula to solve $7 x^2-x=7$, what are the values of $x$?

A. $\frac{1 \pm \sqrt{195} i}{14}$
B. $\frac{1 \pm \sqrt{197}}{14}$
C. $\frac{1 \pm \sqrt{195}}{14}$
D. $\frac{1 \pm \sqrt{197} i}{14}$

Rewrite the equation in standard form: 7 x 2 − x − 7 = 0 .
Identify coefficients: a = 7 , b = − 1 , c = − 7 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c ​ ​ .
Simplify to find the values of x : x = 14 1 ± 197 ​ ​ .

Explanation

Problem Analysis We are given the quadratic equation 7 x 2 − x = 7 . Our goal is to find the values of x that satisfy this equation using the quadratic formula.

Standard Form First, we need to rewrite the equation in the standard form a x 2 + b x + c = 0 . Subtracting 7 from both sides, we get: 7 x 2 − x − 7 = 0

Identifying Coefficients Now, we identify the coefficients a , b , and c . In this case, a = 7 , b = − 1 , and c = − 7 .

Quadratic Formula The quadratic formula is given by: x = 2 a − b ± b 2 − 4 a c ​ ​

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

Simplification Simplify the expression: x = 14 1 ± 1 + 196 ​ ​ x = 14 1 ± 197 ​ ​

Final Answer Therefore, the values of x are 14 1 + 197 ​ ​ and 14 1 − 197 ​ ​ .


Examples
The quadratic formula is a powerful tool used in various fields, such as physics and engineering, to solve problems involving parabolic trajectories or optimizing designs. For example, consider designing a bridge where the supporting cables follow a parabolic path. Using the quadratic formula, engineers can determine the precise points where the cables need to be anchored to ensure structural integrity and stability. This ensures the bridge can withstand different loads and environmental conditions, making it safe and efficient.

Answered by GinnyAnswer | 2025-07-08