Complementary angles

Let $x$ be the angle, then the compliment of $x$ is $90-x$ . Then

$90-x=x^2$

$x^2+x-90=0$

$(x+10)(x-9)=0$

$x=-10$ or $x=9$ (reject the negative value)

So the answer is $x=$ $\boxed{9}$

So basically, the complement of the angle itself would be equal to $90-x$ .

This allows us to start solving with $x ^ 2=90-x$ .

With this information, we can make the quadratic $x^2+x-90=0$ (By adding x and subtracting 90 from both sides).

Assuming the answer is an integer, we can plug this into the quadratic formula to solve for x and get a nice answer. Doing this, we get

$x=\frac{-1 \pm \sqrt{1^2-4(1)(-90)}}{2(1)}=\frac{1 \pm 19}{2}$

This is equal to 9 or -10.

Obviously, x is the measure of the angle and the only possible measure would be the positive one, so here we use $x=9$ . Therefore, 9 is the answer!

Alternatively, you could just recognize that $x ^ 2=90-x$ , then start with the closest perfect square to 90 (81), subtract that from 90, and see if the result for x is equal to its square root. Admittedly, I did none of the above equations, my mind immediately did this second way and I used the equations to check it mathematically before I put 9 as the answer.

according to the question 90-x= x^2

x^2 + x - 90= 0

x (x +10) - 9(x + 10)=0

(x+10)(x-9)=0

therefore x= -10 or 9

since an angle can't be negative.

therefore, x= 9

ans. ) 9 degrees