Solve -x^2+9x+22=0

$$- { x }^{ 2 } +9x+22=0$$

Solve for x

$$x=-2$$

$$x=11$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+22. To find a and b, set up a system to be solved.

$$a+b=9 ab=-22=-22$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -22.

$$-1,22 -2,11$$

Calculate the sum for each pair.

$$-1+22=21 -2+11=9$$

The solution is the pair that gives sum 9.

$$a=11 b=-2$$

Rewrite -x^{2}+9x+22 as \left(-x^{2}+11x\right)+\left(-2x+22\right).

$$\left(-x^{2}+11x\right)+\left(-2x+22\right)$$

Factor out -x in the first and -2 in the second group.

$$-x\left(x-11\right)-2\left(x-11\right)$$

Factor out common term x-11 by using distributive property.

$$\left(x-11\right)\left(-x-2\right)$$

To find equation solutions, solve x-11=0 and -x-2=0.

$$x=11 x=-2$$

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.

$$-x^{2}+9x+22=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 9 for b, and 22 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-1\right)\times 22}}{2\left(-1\right)}$$

Square 9.

$$x=\frac{-9±\sqrt{81-4\left(-1\right)\times 22}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-9±\sqrt{81+4\times 22}}{2\left(-1\right)}$$

Multiply 4 times 22.

$$x=\frac{-9±\sqrt{81+88}}{2\left(-1\right)}$$

Add 81 to 88.

$$x=\frac{-9±\sqrt{169}}{2\left(-1\right)}$$

Take the square root of 169.

$$x=\frac{-9±13}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-9±13}{-2}$$

Now solve the equation x=\frac{-9±13}{-2} when ± is plus. Add -9 to 13.

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

Divide 4 by -2.

$$x=-2$$

Now solve the equation x=\frac{-9±13}{-2} when ± is minus. Subtract 13 from -9.

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

Divide -22 by -2.

$$x=11$$

The equation is now solved.

$$x=-2 x=11$$

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.

$$-x^{2}+9x+22=0$$

Subtract 22 from both sides of the equation.

$$-x^{2}+9x+22-22=-22$$

Subtracting 22 from itself leaves 0.

$$-x^{2}+9x=-22$$

Divide both sides by -1.

$$\frac{-x^{2}+9x}{-1}=-\frac{22}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{9}{-1}x=-\frac{22}{-1}$$

Divide 9 by -1.

$$x^{2}-9x=-\frac{22}{-1}$$

Divide -22 by -1.

$$x^{2}-9x=22$$

Divide -9, the coefficient of the x term, by 2 to get -\frac{9}{2}. Then add the square of -\frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=22+\left(-\frac{9}{2}\right)^{2}$$

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-9x+\frac{81}{4}=22+\frac{81}{4}$$

Add 22 to \frac{81}{4}.

$$x^{2}-9x+\frac{81}{4}=\frac{169}{4}$$

Factor x^{2}-9x+\frac{81}{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}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{169}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{169}{4}}$$

Simplify.

$$x-\frac{9}{2}=\frac{13}{2} x-\frac{9}{2}=-\frac{13}{2}$$

Add \frac{9}{2} to both sides of the equation.

$$x=11 x=-2$$