Solve x^2-4x-3x-9+x^2+3x=0

$${ x }^{ 2 } -4x-3x-9+ { x }^{ 2 } +3x=0$$

Solve for x

$$x=\frac{\sqrt{22}}{2}+1\approx 3.34520788$$

$$x=-\frac{\sqrt{22}}{2}+1\approx -1.34520788$$

Steps Using the Quadratic Formula

Combine -4x and -3x to get -7x.

$$x^{2}-7x-9+x^{2}+3x=0$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}-7x-9+3x=0$$

Combine -7x and 3x to get -4x.

$$2x^{2}-4x-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -4 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 2\left(-9\right)}}{2\times 2}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 2\left(-9\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-4\right)±\sqrt{16-8\left(-9\right)}}{2\times 2}$$

Multiply -8 times -9.

$$x=\frac{-\left(-4\right)±\sqrt{16+72}}{2\times 2}$$

Add 16 to 72.

$$x=\frac{-\left(-4\right)±\sqrt{88}}{2\times 2}$$

Take the square root of 88.

$$x=\frac{-\left(-4\right)±2\sqrt{22}}{2\times 2}$$

The opposite of -4 is 4.

$$x=\frac{4±2\sqrt{22}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{4±2\sqrt{22}}{4}$$

Now solve the equation x=\frac{4±2\sqrt{22}}{4} when ± is plus. Add 4 to 2\sqrt{22}.

$$x=\frac{2\sqrt{22}+4}{4}$$

Divide 4+2\sqrt{22} by 4.

$$x=\frac{\sqrt{22}}{2}+1$$

Now solve the equation x=\frac{4±2\sqrt{22}}{4} when ± is minus. Subtract 2\sqrt{22} from 4.

$$x=\frac{4-2\sqrt{22}}{4}$$

Divide 4-2\sqrt{22} by 4.

$$x=-\frac{\sqrt{22}}{2}+1$$

The equation is now solved.

$$x=\frac{\sqrt{22}}{2}+1 x=-\frac{\sqrt{22}}{2}+1$$

Steps for Completing the Square

Combine -4x and -3x to get -7x.

$$x^{2}-7x-9+x^{2}+3x=0$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}-7x-9+3x=0$$

Combine -7x and 3x to get -4x.

$$2x^{2}-4x-9=0$$

Add 9 to both sides. Anything plus zero gives itself.

$$2x^{2}-4x=9$$

Divide both sides by 2.

$$\frac{2x^{2}-4x}{2}=\frac{9}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{4}{2}\right)x=\frac{9}{2}$$

Divide -4 by 2.

$$x^{2}-2x=\frac{9}{2}$$

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.

$$x^{2}-2x+1=\frac{9}{2}+1$$

Add \frac{9}{2} to 1.

$$x^{2}-2x+1=\frac{11}{2}$$

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}.

$$\left(x-1\right)^{2}=\frac{11}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{11}{2}}$$

Simplify.

$$x-1=\frac{\sqrt{22}}{2} x-1=-\frac{\sqrt{22}}{2}$$

Add 1 to both sides of the equation.

$$x=\frac{\sqrt{22}}{2}+1 x=-\frac{\sqrt{22}}{2}+1$$