Solve x=x^2+4x-5x-20

$$x = x ^ { 2 } + 4 x - 5 x - 20$$

Solve for x

$$x=\sqrt{21}+1\approx 5.582575695$$

$$x=1-\sqrt{21}\approx -3.582575695$$

Steps Using the Quadratic Formula

Combine 4x and -5x to get -x.

$$x=x^{2}-x-20$$

Subtract x^{2} from both sides.

$$x-x^{2}=-x-20$$

Add x to both sides.

$$x-x^{2}+x=-20$$

Combine x and x to get 2x.

$$2x-x^{2}=-20$$

Add 20 to both sides.

$$2x-x^{2}+20=0$$

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.

$$-x^{2}+2x+20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 2 for b, and 20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-1\right)\times 20}}{2\left(-1\right)}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-1\right)\times 20}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-2±\sqrt{4+4\times 20}}{2\left(-1\right)}$$

Multiply 4 times 20.

$$x=\frac{-2±\sqrt{4+80}}{2\left(-1\right)}$$

Add 4 to 80.

$$x=\frac{-2±\sqrt{84}}{2\left(-1\right)}$$

Take the square root of 84.

$$x=\frac{-2±2\sqrt{21}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-2±2\sqrt{21}}{-2}$$

Now solve the equation x=\frac{-2±2\sqrt{21}}{-2} when ± is plus. Add -2 to 2\sqrt{21}.

$$x=\frac{2\sqrt{21}-2}{-2}$$

Divide -2+2\sqrt{21} by -2.

$$x=1-\sqrt{21}$$

Now solve the equation x=\frac{-2±2\sqrt{21}}{-2} when ± is minus. Subtract 2\sqrt{21} from -2.

$$x=\frac{-2\sqrt{21}-2}{-2}$$

Divide -2-2\sqrt{21} by -2.

$$x=\sqrt{21}+1$$

The equation is now solved.

$$x=1-\sqrt{21} x=\sqrt{21}+1$$

Steps for Completing the Square

Combine 4x and -5x to get -x.

$$x=x^{2}-x-20$$

Subtract x^{2} from both sides.

$$x-x^{2}=-x-20$$

Add x to both sides.

$$x-x^{2}+x=-20$$

Combine x and x to get 2x.

$$2x-x^{2}=-20$$

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.

$$-x^{2}+2x=-20$$

Divide both sides by -1.

$$\frac{-x^{2}+2x}{-1}=-\frac{20}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{2}{-1}x=-\frac{20}{-1}$$

Divide 2 by -1.

$$x^{2}-2x=-\frac{20}{-1}$$

Divide -20 by -1.

$$x^{2}-2x=20$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=20+1$$

Add 20 to 1.

$$x^{2}-2x+1=21$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=21$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{21}$$

Simplify.

$$x-1=\sqrt{21} x-1=-\sqrt{21}$$

Add 1 to both sides of the equation.

$$x=\sqrt{21}+1 x=1-\sqrt{21}$$