Solve x^2+3x=0x+20+x

$$x ^ { 2 } + 3 x = 0 x + 20 + x$$

Solve for x (complex solution)

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

$$x=-\left(\sqrt{21}+1\right)\approx -5.582575695$$

Steps Using the Quadratic Formula

Anything times zero gives zero.

$$x^{2}+3x=0+20+x$$

Add 0 and 20 to get 20.

$$x^{2}+3x=20+x$$

Subtract 20 from both sides.

$$x^{2}+3x-20=x$$

Subtract x from both sides.

$$x^{2}+3x-20-x=0$$

Combine 3x and -x to get 2x.

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

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}.

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

Square 2.

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

Multiply -4 times -20.

$$x=\frac{-2±\sqrt{4+80}}{2}$$

Add 4 to 80.

$$x=\frac{-2±\sqrt{84}}{2}$$

Take the square root of 84.

$$x=\frac{-2±2\sqrt{21}}{2}$$

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.

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

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.

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

Steps for Completing the Square

Anything times zero gives zero.

$$x^{2}+3x=0+20+x$$

Add 0 and 20 to get 20.

$$x^{2}+3x=20+x$$

Subtract x from both sides.

$$x^{2}+3x-x=20$$

Combine 3x and -x to get 2x.

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

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.

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

Square 1.

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

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.

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

Subtract 1 from both sides of the equation.

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

Solve for x

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

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

Steps Using the Quadratic Formula

Anything times zero gives zero.

$$x^{2}+3x=0+20+x$$

Add 0 and 20 to get 20.

$$x^{2}+3x=20+x$$

Subtract 20 from both sides.

$$x^{2}+3x-20=x$$

Subtract x from both sides.

$$x^{2}+3x-20-x=0$$

Combine 3x and -x to get 2x.

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

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}.

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

Square 2.

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

Multiply -4 times -20.

$$x=\frac{-2±\sqrt{4+80}}{2}$$

Add 4 to 80.

$$x=\frac{-2±\sqrt{84}}{2}$$

Take the square root of 84.

$$x=\frac{-2±2\sqrt{21}}{2}$$

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.

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

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.

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

Steps for Completing the Square

Anything times zero gives zero.

$$x^{2}+3x=0+20+x$$

Add 0 and 20 to get 20.

$$x^{2}+3x=20+x$$

Subtract x from both sides.

$$x^{2}+3x-x=20$$

Combine 3x and -x to get 2x.

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

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.

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

Square 1.

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

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.

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

Subtract 1 from both sides of the equation.

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