Solve for $x$ in the equation $x^2-12 x+59=0$
A. $x=-12 \pm \sqrt{85}$
B. $x=-6 \pm \sqrt{23} i$
C. $x=6 \pm \sqrt{23} i$
D. $x=12 \pm \sqrt{85}$

Apply the quadratic formula x = 2 a − b ± b 2 − 4 a c ​ ​ with a = 1 , b = − 12 , and c = 59 .
Simplify the expression to x = 2 12 ± − 92 ​ ​ .
Rewrite − 92 ​ as 2 23 ​ i .
Simplify further to obtain the final answer: x = 6 ± 23 ​ i .

Explanation

Understanding the Problem We are given the quadratic equation x 2 − 12 x + 59 = 0 . Our goal is to find the values of x that satisfy this equation.

Applying the Quadratic Formula We can use the quadratic formula to solve for x . The quadratic formula is given by: x = 2 a − b ± b 2 − 4 a c ​ ​ where a , b , and c are the coefficients of the quadratic equation a x 2 + b x + c = 0 .

Substituting the Values In our equation, x 2 − 12 x + 59 = 0 , we have a = 1 , b = − 12 , and c = 59 . Substituting these values into the quadratic formula, we get: x = 2 ( 1 ) − ( − 12 ) ± ( − 12 ) 2 − 4 ( 1 ) ( 59 ) ​ ​

Simplifying the Expression Now, let's simplify the expression: x = 2 12 ± 144 − 236 ​ ​ x = 2 12 ± − 92 ​ ​

Dealing with the Negative Discriminant Since the discriminant (the value inside the square root) is negative, we will have complex solutions. We can rewrite − 92 ​ as 92 ​ i . Furthermore, we can simplify 92 ​ as 4 ⋅ 23 ​ = 2 23 ​ . Therefore, − 92 ​ = 2 23 ​ i .

Final Simplification Substituting this back into our equation for x , we get: x = 2 12 ± 2 23 ​ i ​ Now, we can divide both terms in the numerator by 2: x = 6 ± 23 ​ i

Stating the Solutions Thus, the solutions for x are 6 + 23 ​ i and 6 − 23 ​ i .


Examples
Quadratic equations are not just abstract math; they appear in various real-world applications. For instance, when designing a parabolic reflector for a flashlight or satellite dish, engineers use quadratic equations to determine the optimal shape that focuses light or radio waves efficiently. Similarly, in physics, projectile motion, such as the trajectory of a ball thrown in the air, can be modeled using quadratic equations to predict its range and maximum height. Understanding how to solve these equations allows us to make accurate predictions and design effective technologies.

Answered by GinnyAnswer | 2025-07-03

The solutions for the equation x 2 − 12 x + 59 = 0 are 6 + 23 ​ i and 6 − 23 ​ i , leading us to choose option C: x = 6 ± 23 ​ i .
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Answered by Anonymous | 2025-07-04