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)
To solve the equation x 2 − 12 x + 36 = 90 , we rearranged it to x 2 − 12 x − 54 = 0 and applied the quadratic formula. After simplifying, we arrived at the solution x = 6 ± 3 10 , which is option A. Thus, the answer is A.
;
To solve the equation x 2 − 12 x + 36 = 90 for x , we start by simplifying and rearranging the equation.
First, subtract 90 from both sides to set the equation to zero:
x 2 − 12 x + 36 − 90 = 0
This simplifies to:
x 2 − 12 x − 54 = 0
Next, we need to solve this quadratic equation. We can use the quadratic formula, which is given by:
x = 2 a − b ± b 2 − 4 a c
Here, a = 1 , b = − 12 , and c = − 54 . Plug these values into the quadratic formula:
x = 2 ⋅ 1 − ( − 12 ) ± ( − 12 ) 2 − 4 ⋅ 1 ⋅ ( − 54 )
Simplify the expression:
x = 2 12 ± 144 + 216
x = 2 12 ± 360
At this point, notice that 360 = 36 ⋅ 10 = 6 10 . Thus:
x = 2 12 ± 6 10
Now, divide each term by 2:
x = 6 ± 3 10
Therefore, the solution to the equation is x = 6 ± 3 10 .
The correct multiple-choice answer is A. x = 6 ± 3 10 .
