Solve +(x)=x^2-36x

$$+ ( x ) = x ^ { 2 } - 36 x$$

Solve for x

$$x=37$$

$$x=0$$

Steps Using Factoring

Subtract x^{2} from both sides.

$$x-x^{2}=-36x$$

Add 36x to both sides.

$$x-x^{2}+36x=0$$

Combine x and 36x to get 37x.

$$37x-x^{2}=0$$

Factor out x.

$$x\left(37-x\right)=0$$

To find equation solutions, solve x=0 and 37-x=0.

$$x=0 x=37$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

$$x-x^{2}=-36x$$

Add 36x to both sides.

$$x-x^{2}+36x=0$$

Combine x and 36x to get 37x.

$$37x-x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-x^{2}+37x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 37 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-37±\sqrt{37^{2}}}{2\left(-1\right)}$$

Take the square root of 37^{2}.

$$x=\frac{-37±37}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-37±37}{-2}$$

Now solve the equation x=\frac{-37±37}{-2} when ± is plus. Add -37 to 37.

$$x=\frac{0}{-2}$$

Divide 0 by -2.

$$x=0$$

Now solve the equation x=\frac{-37±37}{-2} when ± is minus. Subtract 37 from -37.

$$x=-\frac{74}{-2}$$

Divide -74 by -2.

$$x=37$$

The equation is now solved.

$$x=0 x=37$$

Steps for Completing the Square

Subtract x^{2} from both sides.

$$x-x^{2}=-36x$$

Add 36x to both sides.

$$x-x^{2}+36x=0$$

Combine x and 36x to get 37x.

$$37x-x^{2}=0$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-x^{2}+37x=0$$

Divide both sides by -1.

$$\frac{-x^{2}+37x}{-1}=\frac{0}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{37}{-1}x=\frac{0}{-1}$$

Divide 37 by -1.

$$x^{2}-37x=\frac{0}{-1}$$

Divide 0 by -1.

$$x^{2}-37x=0$$

Divide -37, the coefficient of the x term, by 2 to get -\frac{37}{2}. Then add the square of -\frac{37}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-37x+\left(-\frac{37}{2}\right)^{2}=\left(-\frac{37}{2}\right)^{2}$$

Square -\frac{37}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-37x+\frac{1369}{4}=\frac{1369}{4}$$

Factor x^{2}-37x+\frac{1369}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{37}{2}\right)^{2}=\frac{1369}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{37}{2}\right)^{2}}=\sqrt{\frac{1369}{4}}$$

Simplify.

$$x-\frac{37}{2}=\frac{37}{2} x-\frac{37}{2}=-\frac{37}{2}$$

Add \frac{37}{2} to both sides of the equation.

$$x=37 x=0$$