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 as ( x − 6 ) 2 = 90 .
Take the square root of both sides: x − 6 = ± 90 .
Simplify the square root: 90 = 3 10 .
Solve for x : x = 6 ± 3 10 . The solution is 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 . Notice that the left side of the equation is a perfect square.
Rewriting the Equation We can rewrite the equation as ( x − 6 ) 2 = 90 . This simplifies the equation and makes it easier to solve.
Taking the Square Root Now, we take the square root of both sides of the equation: x − 6 = ± 90 . Remember to include both the positive and negative square roots.
Simplifying the Square Root Next, we simplify the square root: 90 = 9 ⋅ 10 = 9 ⋅ 10 = 3 10 .
Solving for x Finally, we solve for x by adding 6 to both sides: x = 6 ± 3 10 .
Final Answer Therefore, the solutions for x are x = 6 + 3 10 and x = 6 − 3 10 .
Examples
Imagine you are designing a square garden and want to increase its area by a certain amount. This problem is similar to finding the new side length of the garden after increasing its area. Understanding how to solve quadratic equations helps in various real-world scenarios, such as calculating areas, designing structures, and optimizing processes. This algebraic approach ensures accurate and efficient solutions in practical tasks.
The solution to the equation x 2 − 12 x + 36 = 90 is x = 6 ± 3 10 . The simplification involves recognizing the left side as a perfect square and solving for x after taking the square root. The correct choice is option A: x = 6 ± 3 10 .
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