Solve x^2+7x=4x+20

$$x ^ { 2 } + 7 x = 4 x + 20$$

Solve for x

$$x = \frac{\sqrt{89} - 3}{2} \approx 3.216990566$$

$$x=\frac{-\sqrt{89}-3}{2}\approx -6.216990566$$

Steps Using the Quadratic Formula

Subtract 4x from both sides.

$$x^{2}+7x-4x=20$$

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

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

Subtract 20 from both sides.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 3 for b, and -20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-20\right)}}{2}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\left(-20\right)}}{2}$$

Multiply -4 times -20.

$$x=\frac{-3±\sqrt{9+80}}{2}$$

Add 9 to 80.

$$x=\frac{-3±\sqrt{89}}{2}$$

Now solve the equation x=\frac{-3±\sqrt{89}}{2} when ± is plus. Add -3 to \sqrt{89}.

$$x=\frac{\sqrt{89}-3}{2}$$

Now solve the equation x=\frac{-3±\sqrt{89}}{2} when ± is minus. Subtract \sqrt{89} from -3.

$$x=\frac{-\sqrt{89}-3}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{89}-3}{2} x=\frac{-\sqrt{89}-3}{2}$$

Steps for Completing the Square

Subtract 4x from both sides.

$$x^{2}+7x-4x=20$$

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

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

Divide 3, the coefficient of the x term, by 2 to get \frac{3}{2}. Then add the square of \frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=20+\left(\frac{3}{2}\right)^{2}$$

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+3x+\frac{9}{4}=20+\frac{9}{4}$$

Add 20 to \frac{9}{4}.

$$x^{2}+3x+\frac{9}{4}=\frac{89}{4}$$

Factor x^{2}+3x+\frac{9}{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}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{89}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{89}{4}}$$

Simplify.

$$x+\frac{3}{2}=\frac{\sqrt{89}}{2} x+\frac{3}{2}=-\frac{\sqrt{89}}{2}$$

Subtract \frac{3}{2} from both sides of the equation.

$$x=\frac{\sqrt{89}-3}{2} x=\frac{-\sqrt{89}-3}{2}$$