Solve x^2-16-x=8x+6

$$x ^ { 2 } - 16 - x = 8 x + 6$$

Solve for x

$$x=-2$$

$$x=11$$

Steps Using Factoring

Subtract 8x from both sides.

$$x^{2}-16-x-8x=6$$

Combine -x and -8x to get -9x.

$$x^{2}-16-9x=6$$

Subtract 6 from both sides.

$$x^{2}-16-9x-6=0$$

Subtract 6 from -16 to get -22.

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

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

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

To solve the equation, factor x^{2}-9x-22 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

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

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. 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 factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

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

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

$$x=11 x=-2$$

Steps Using Factoring By Grouping

Subtract 8x from both sides.

$$x^{2}-16-x-8x=6$$

Combine -x and -8x to get -9x.

$$x^{2}-16-9x=6$$

Subtract 6 from both sides.

$$x^{2}-16-9x-6=0$$

Subtract 6 from -16 to get -22.

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

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

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

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.

$$a+b=-9 ab=1\left(-22\right)=-22$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. 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

Subtract 8x from both sides.

$$x^{2}-16-x-8x=6$$

Combine -x and -8x to get -9x.

$$x^{2}-16-9x=6$$

Subtract 6 from both sides.

$$x^{2}-16-9x-6=0$$

Subtract 6 from -16 to get -22.

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

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}.

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

Square -9.

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

Multiply -4 times -22.

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

Add 81 to 88.

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

Take the square root of 169.

$$x=\frac{-\left(-9\right)±13}{2}$$

The opposite of -9 is 9.

$$x=\frac{9±13}{2}$$

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

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

Divide 22 by 2.

$$x=11$$

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

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

Divide -4 by 2.

$$x=-2$$

The equation is now solved.

$$x=11 x=-2$$

Steps for Completing the Square

Subtract 8x from both sides.

$$x^{2}-16-x-8x=6$$

Combine -x and -8x to get -9x.

$$x^{2}-16-9x=6$$

Add 16 to both sides.

$$x^{2}-9x=6+16$$

Add 6 and 16 to get 22.

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

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.

$$x=11 x=-2$$