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 (1)
Recognize the left side of the equation as a perfect square: ( x − 6 ) 2 = 90 .
Take the square root of both sides: x − 6 = ± 90 .
Simplify the radical: 90 = 3 10 .
Solve for x : x = 6 ± 3 10 .
Explanation
Recognizing Perfect Square We are given the equation x 2 − 12 x + 36 = 90 . Notice that the left side is a perfect square trinomial. We can rewrite the equation as ( x − 6 ) 2 = 90 .
Taking Square Root Now, we take the square root of both sides of the equation: ( x − 6 ) 2 = ± 90 . This simplifies to x − 6 = ± 90 .
Simplifying the Radical We can simplify 90 by factoring out the largest perfect square factor, which is 9. So, 90 = 9 × 10 = 9 × 10 = 3 10 .
Solving for x Substituting this back into our equation, we have x − 6 = ± 3 10 . 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 with side length x . You want to add a walkway around the garden that is 3 feet wide. The total area of the garden and walkway is 90 square feet. The original garden's area is x 2 . The area of the garden plus the walkway can be represented by ( x + 6 ) 2 = 90 , where 6 represents the increase in side length due to the 3-foot walkway on each side. Solving this equation for x will tell you the original side length of the garden.
