Solve x^2+16x=30x+90

$$x ^ { 2 } + 16 x = 30 x + 90$$

Solve for x

$$x=\sqrt{139}+7\approx 18.789826123$$

$$x=7-\sqrt{139}\approx -4.789826123$$

Steps Using the Quadratic Formula

Subtract 30x from both sides.

$$x^{2}+16x-30x=90$$

Combine 16x and -30x to get -14x.

$$x^{2}-14x=90$$

Subtract 90 from both sides.

$$x^{2}-14x-90=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -14 for b, and -90 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\left(-90\right)}}{2}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\left(-90\right)}}{2}$$

Multiply -4 times -90.

$$x=\frac{-\left(-14\right)±\sqrt{196+360}}{2}$$

Add 196 to 360.

$$x=\frac{-\left(-14\right)±\sqrt{556}}{2}$$

Take the square root of 556.

$$x=\frac{-\left(-14\right)±2\sqrt{139}}{2}$$

The opposite of -14 is 14.

$$x=\frac{14±2\sqrt{139}}{2}$$

Now solve the equation x=\frac{14±2\sqrt{139}}{2} when ± is plus. Add 14 to 2\sqrt{139}.

$$x=\frac{2\sqrt{139}+14}{2}$$

Divide 14+2\sqrt{139} by 2.

$$x=\sqrt{139}+7$$

Now solve the equation x=\frac{14±2\sqrt{139}}{2} when ± is minus. Subtract 2\sqrt{139} from 14.

$$x=\frac{14-2\sqrt{139}}{2}$$

Divide 14-2\sqrt{139} by 2.

$$x=7-\sqrt{139}$$

The equation is now solved.

$$x=\sqrt{139}+7 x=7-\sqrt{139}$$

Steps for Completing the Square

Subtract 30x from both sides.

$$x^{2}+16x-30x=90$$

Combine 16x and -30x to get -14x.

$$x^{2}-14x=90$$

Divide -14, the coefficient of the x term, by 2 to get -7. Then add the square of -7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-14x+\left(-7\right)^{2}=90+\left(-7\right)^{2}$$

Square -7.

$$x^{2}-14x+49=90+49$$

Add 90 to 49.

$$x^{2}-14x+49=139$$

Factor x^{2}-14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-7\right)^{2}=139$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{139}$$

Simplify.

$$x-7=\sqrt{139} x-7=-\sqrt{139}$$

Add 7 to both sides of the equation.

$$x=\sqrt{139}+7 x=7-\sqrt{139}$$