Using the quadratic formula to solve $4 x^2-3 x+9=2 x+1$, what are the values of $x$?

A. $\frac{1 \pm \sqrt{159}}{8}$
B. $\frac{5 \pm \sqrt{153}}{8}$
C. $\frac{5 \pm \sqrt{103 i}}{8}$
D. $\frac{1 \pm \sqrt{153}}{8}$

The equation 4 x 2 − 3 x + 9 = 2 x + 1 is rewritten as 4 x 2 − 5 x + 8 = 0 and solved using the quadratic formula, revealing complex roots. The final solutions are x = 8 5 ± 103 ​ i ​ , making option C the correct choice.
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Answered by Anonymous | 2025-07-04

Rewrite the equation in standard quadratic form: 4 x 2 − 5 x + 8 = 0 .
Identify the coefficients: a = 4 , b = − 5 , c = 8 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c ​ ​ = 8 5 ± − 103 ​ ​ .
Simplify to find the complex roots: x = 8 5 ± 103 ​ i ​ .

Explanation

Rewrite the equation First, we need to rewrite the given equation 4 x 2 − 3 x + 9 = 2 x + 1 in the standard quadratic form a x 2 + b x + c = 0 . To do this, subtract 2 x and 1 from both sides of the equation: 4 x 2 − 3 x + 9 − 2 x − 1 = 0
4 x 2 − 5 x + 8 = 0

Identify the coefficients Now, we can identify the coefficients a , b , and c in the quadratic equation 4 x 2 − 5 x + 8 = 0 . We have: a = 4 b = − 5 c = 8

Apply the quadratic formula Next, we apply the quadratic formula, which is given by: x = 2 a − b ± b 2 − 4 a c ​ ​ Substitute the values of a , b , and c into the formula: x = 2 ( 4 ) − ( − 5 ) ± ( − 5 ) 2 − 4 ( 4 ) ( 8 ) ​ ​ x = 8 5 ± 25 − 128 ​ ​ x = 8 5 ± − 103 ​ ​

Simplify the expression Since the discriminant is negative, we have complex roots. We can rewrite the expression as: x = 8 5 ± 103 × − 1 ​ ​ x = 8 5 ± 103 ​ i ​ So the values of x are: x = 8 5 + 103 ​ i ​ and x = 8 5 − 103 ​ i ​

Final Answer Therefore, the values of x are 8 5 ± 103 ​ i ​ .


Examples
Quadratic equations are incredibly useful in physics, engineering, and even economics. For example, when designing a bridge, engineers use quadratic equations to calculate the curve of suspension cables. Similarly, in finance, quadratic equations can help model investment growth or calculate break-even points for businesses. Understanding how to solve these equations allows professionals to make informed decisions and create efficient designs.

Answered by GinnyAnswer | 2025-07-04