Solve for $x$ in the equation $x^2-12 x+36=90$.
A. $x=6 \pm 3 \sqrt{10}$
B. $x=6 \pm 2 \sqrt{7}$
C. $x=12 \pm 3 \sqrt{22}$
D. $x=12 \pm 3 \sqrt{10}$
Answer (2)
Rewrite the equation in the standard quadratic form: x 2 − 12 x − 54 = 0 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c , where a = 1 , b = − 12 , and c = − 54 .
Simplify the expression to find the solutions: x = 6 ± 3 10 .
The solutions for x are: x = 6 ± 3 10 .
Explanation
Understanding the Problem We are given the quadratic equation x 2 − 12 x + 36 = 90 . Our goal is to solve for x .
Rewriting the Equation First, let's rewrite the equation by subtracting 90 from both sides: x 2 − 12 x + 36 − 90 = 0 . This simplifies to x 2 − 12 x − 54 = 0 .
Applying the Quadratic Formula Now, 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 = 1 , b = − 12 , and c = − 54 .
Substituting Values Substitute the values of a , b , and c into the quadratic formula: x = 2 ( 1 ) − ( − 12 ) ± ( − 12 ) 2 − 4 ( 1 ) ( − 54 ) .
Simplifying the Expression Simplify the expression: x = 2 12 ± 144 + 216 = 2 12 ± 360 .
Simplifying the Square Root Now, let's simplify the square root: 360 = 36 ⋅ 10 = 6 10 .
Substituting Back Substitute this back into the equation: x = 2 12 ± 6 10 .
Final Solution Finally, divide both terms in the numerator by 2: x = 6 ± 3 10 . Therefore, the solutions are x = 6 + 3 10 and x = 6 − 3 10 .
Examples
Imagine you are designing a rectangular garden where the area is determined by the equation x 2 − 12 x + 36 = 90 , where x represents a dimension of the garden. Solving this quadratic equation helps you find the possible values for x , which in turn helps you determine the dimensions of the garden. Understanding how to solve such equations is crucial in various fields like engineering, physics, and economics, where quadratic relationships often arise in modeling real-world scenarios. This algebraic skill enables you to optimize designs and make informed decisions based on mathematical models.
To solve the equation x 2 − 12 x + 36 = 90 , we rearranged it to the standard quadratic form x 2 − 12 x − 54 = 0 and applied the quadratic formula. The solutions simplify to x = 6 ± 3 10 , making option A the correct answer. The final solutions are x = 6 + 3 10 and x = 6 − 3 10 .
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