Solve the equation for x.
[tex]$3 x^2=8 x-7$[/tex]
A) [tex]$\left{\frac{4 \pm i \sqrt{5}}{6}\right}$[/tex]

Rewrite the equation in standard form: 3 x 2 − 8 x + 7 = 0 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c ​ ​ .
Substitute a = 3 , b = − 8 , and c = 7 into the formula and simplify.
The solutions are x = 3 4 ± i 5 ​ ​ , so the answer is 3 4 ± i 5 ​ ​ ​ .

Explanation

Problem Analysis We are given the quadratic equation 3 x 2 = 8 x − 7 . Our goal is to solve for x .

Rewriting the Equation First, we rewrite the equation in the standard quadratic form a x 2 + b x + c = 0 . Subtracting 8 x and adding 7 to both sides, we get: 3 x 2 − 8 x + 7 = 0

Identifying Coefficients Now, we identify the coefficients: a = 3 , b = − 8 , and c = 7 .

Applying the Quadratic Formula We apply the quadratic formula to find the solutions for x : x = 2 a − b ± b 2 − 4 a c ​ ​

Substituting Values Substitute the values of a , b , and c into the formula: x = 2 ( 3 ) − ( − 8 ) ± ( − 8 ) 2 − 4 ( 3 ) ( 7 ) ​ ​ x = 6 8 ± 64 − 84 ​ ​ x = 6 8 ± − 20 ​ ​

Simplifying the Square Root Since the discriminant is negative, we have complex solutions. We simplify the square root of − 20 : − 20 ​ = 20 ​ ⋅ − 1 ​ = 4 ⋅ 5 ​ ⋅ i = 2 i 5 ​

Simplifying the Solution Substitute this back into the expression for x : x = 6 8 ± 2 i 5 ​ ​ x = 2 ( 3 ) 2 ( 4 ± i 5 ​ ) ​ x = 3 4 ± i 5 ​ ​

Final Solutions Therefore, the solutions are: x = 3 4 + i 5 ​ ​ , x = 3 4 − i 5 ​ ​


Examples
Quadratic equations are used in various fields such as physics, engineering, and economics. For example, in physics, they can be used to model the trajectory of a projectile. In finance, they can be used to model cost and revenue to determine break-even points.

Answered by GinnyAnswer | 2025-07-03

To solve the equation 3 x 2 = 8 x − 7 , we rewrite it as 3 x 2 − 8 x + 7 = 0 and then apply the quadratic formula. The solutions are complex numbers given by x = 3 4 ± i 5 ​ ​ . Thus, the answer is 3 4 ± i 5 ​ ​ .
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Answered by Anonymous | 2025-07-04