Solve x^2+19x+100=9648

$${ x }^{ 2 } +19x+100=9648$$

Solve for x

$$x = \frac{\sqrt{38553} - 19}{2} \approx 88.67458938$$

$$x=\frac{-\sqrt{38553}-19}{2}\approx -107.67458938$$

Steps Using the Quadratic Formula

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}+19x+100=9648$$

Subtract 9648 from both sides of the equation.

$$x^{2}+19x+100-9648=9648-9648$$

Subtracting 9648 from itself leaves 0.

$$x^{2}+19x+100-9648=0$$

Subtract 9648 from 100.

$$x^{2}+19x-9548=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 19 for b, and -9548 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-19±\sqrt{19^{2}-4\left(-9548\right)}}{2}$$

Square 19.

$$x=\frac{-19±\sqrt{361-4\left(-9548\right)}}{2}$$

Multiply -4 times -9548.

$$x=\frac{-19±\sqrt{361+38192}}{2}$$

Add 361 to 38192.

$$x=\frac{-19±\sqrt{38553}}{2}$$

Now solve the equation x=\frac{-19±\sqrt{38553}}{2} when ± is plus. Add -19 to \sqrt{38553}.

$$x=\frac{\sqrt{38553}-19}{2}$$

Now solve the equation x=\frac{-19±\sqrt{38553}}{2} when ± is minus. Subtract \sqrt{38553} from -19.

$$x=\frac{-\sqrt{38553}-19}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{38553}-19}{2} x=\frac{-\sqrt{38553}-19}{2}$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}+19x+100=9648$$

Subtract 100 from both sides of the equation.

$$x^{2}+19x+100-100=9648-100$$

Subtracting 100 from itself leaves 0.

$$x^{2}+19x=9648-100$$

Subtract 100 from 9648.

$$x^{2}+19x=9548$$

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.

$$x^{2}+19x+\left(\frac{19}{2}\right)^{2}=9548+\left(\frac{19}{2}\right)^{2}$$

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

$$x^{2}+19x+\frac{361}{4}=9548+\frac{361}{4}$$

Add 9548 to \frac{361}{4}.

$$x^{2}+19x+\frac{361}{4}=\frac{38553}{4}$$

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}.

$$\left(x+\frac{19}{2}\right)^{2}=\frac{38553}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{19}{2}\right)^{2}}=\sqrt{\frac{38553}{4}}$$

Simplify.

$$x+\frac{19}{2}=\frac{\sqrt{38553}}{2} x+\frac{19}{2}=-\frac{\sqrt{38553}}{2}$$

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

$$x=\frac{\sqrt{38553}-19}{2} x=\frac{-\sqrt{38553}-19}{2}$$