Solve (x+19)x=8100

$$(x+19)x=8100$$

Solve for x

$$x=-100$$

$$x=81$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x+19 by x.

$$x^{2}+19x=8100$$

Subtract 8100 from both sides.

$$x^{2}+19x-8100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 19 for b, and -8100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-19±\sqrt{19^{2}-4\left(-8100\right)}}{2}$$

Square 19.

$$x=\frac{-19±\sqrt{361-4\left(-8100\right)}}{2}$$

Multiply -4 times -8100.

$$x=\frac{-19±\sqrt{361+32400}}{2}$$

Add 361 to 32400.

$$x=\frac{-19±\sqrt{32761}}{2}$$

Take the square root of 32761.

$$x=\frac{-19±181}{2}$$

Now solve the equation x=\frac{-19±181}{2} when ± is plus. Add -19 to 181.

$$x=\frac{162}{2}$$

Divide 162 by 2.

$$x=81$$

Now solve the equation x=\frac{-19±181}{2} when ± is minus. Subtract 181 from -19.

$$x=-\frac{200}{2}$$

Divide -200 by 2.

$$x=-100$$

The equation is now solved.

$$x=81 x=-100$$

Steps for Completing the Square

Use the distributive property to multiply x+19 by x.

$$x^{2}+19x=8100$$

Divide 19, the coefficient of the x term, by 2 to get \frac{19}{2}. Then add the square of \frac{19}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+19x+\left(\frac{19}{2}\right)^{2}=8100+\left(\frac{19}{2}\right)^{2}$$

Square \frac{19}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+19x+\frac{361}{4}=8100+\frac{361}{4}$$

Add 8100 to \frac{361}{4}.

$$x^{2}+19x+\frac{361}{4}=\frac{32761}{4}$$

Factor x^{2}+19x+\frac{361}{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}.

$$\left(x+\frac{19}{2}\right)^{2}=\frac{32761}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{19}{2}\right)^{2}}=\sqrt{\frac{32761}{4}}$$

Simplify.

$$x+\frac{19}{2}=\frac{181}{2} x+\frac{19}{2}=-\frac{181}{2}$$

Subtract \frac{19}{2} from both sides of the equation.

$$x=81 x=-100$$