Solve x^2-8x-48=(x-12)

$$x ^ { 2 } - 8 x - 48 = ( x - 12 )$$

Solve for x

$$x=-3$$

$$x=12$$

Steps Using Factoring

Subtract x from both sides.

$$x^{2}-8x-48-x=-12$$

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

$$x^{2}-9x-48=-12$$

Add 12 to both sides.

$$x^{2}-9x-48+12=0$$

Add -48 and 12 to get -36.

$$x^{2}-9x-36=0$$

To solve the equation, factor x^{2}-9x-36 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.

$$a+b=-9 ab=-36$$

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 -36.

$$1,-36 2,-18 3,-12 4,-9 6,-6$$

Calculate the sum for each pair.

$$1-36=-35 2-18=-16 3-12=-9 4-9=-5 6-6=0$$

The solution is the pair that gives sum -9.

$$a=-12 b=3$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-12\right)\left(x+3\right)$$

To find equation solutions, solve x-12=0 and x+3=0.

$$x=12 x=-3$$

Steps Using Factoring By Grouping

Subtract x from both sides.

$$x^{2}-8x-48-x=-12$$

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

$$x^{2}-9x-48=-12$$

Add 12 to both sides.

$$x^{2}-9x-48+12=0$$

Add -48 and 12 to get -36.

$$x^{2}-9x-36=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-36. To find a and b, set up a system to be solved.

$$a+b=-9 ab=1\left(-36\right)=-36$$

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 -36.

$$1,-36 2,-18 3,-12 4,-9 6,-6$$

Calculate the sum for each pair.

$$1-36=-35 2-18=-16 3-12=-9 4-9=-5 6-6=0$$

The solution is the pair that gives sum -9.

$$a=-12 b=3$$

Rewrite x^{2}-9x-36 as \left(x^{2}-12x\right)+\left(3x-36\right).

$$\left(x^{2}-12x\right)+\left(3x-36\right)$$

Factor out x in the first and 3 in the second group.

$$x\left(x-12\right)+3\left(x-12\right)$$

Factor out common term x-12 by using distributive property.

$$\left(x-12\right)\left(x+3\right)$$

To find equation solutions, solve x-12=0 and x+3=0.

$$x=12 x=-3$$

Steps Using the Quadratic Formula

Subtract x from both sides.

$$x^{2}-8x-48-x=-12$$

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

$$x^{2}-9x-48=-12$$

Add 12 to both sides.

$$x^{2}-9x-48+12=0$$

Add -48 and 12 to get -36.

$$x^{2}-9x-36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -9 for b, and -36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-36\right)}}{2}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\left(-36\right)}}{2}$$

Multiply -4 times -36.

$$x=\frac{-\left(-9\right)±\sqrt{81+144}}{2}$$

Add 81 to 144.

$$x=\frac{-\left(-9\right)±\sqrt{225}}{2}$$

Take the square root of 225.

$$x=\frac{-\left(-9\right)±15}{2}$$

The opposite of -9 is 9.

$$x=\frac{9±15}{2}$$

Now solve the equation x=\frac{9±15}{2} when ± is plus. Add 9 to 15.

$$x=\frac{24}{2}$$

Divide 24 by 2.

$$x=12$$

Now solve the equation x=\frac{9±15}{2} when ± is minus. Subtract 15 from 9.

$$x=-\frac{6}{2}$$

Divide -6 by 2.

$$x=-3$$

The equation is now solved.

$$x=12 x=-3$$

Steps for Completing the Square

Subtract x from both sides.

$$x^{2}-8x-48-x=-12$$

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

$$x^{2}-9x-48=-12$$

Add 48 to both sides.

$$x^{2}-9x=-12+48$$

Add -12 and 48 to get 36.

$$x^{2}-9x=36$$

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.

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=36+\left(-\frac{9}{2}\right)^{2}$$

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

$$x^{2}-9x+\frac{81}{4}=36+\frac{81}{4}$$

Add 36 to \frac{81}{4}.

$$x^{2}-9x+\frac{81}{4}=\frac{225}{4}$$

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}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{225}{4}}$$

Simplify.

$$x-\frac{9}{2}=\frac{15}{2} x-\frac{9}{2}=-\frac{15}{2}$$

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

$$x=12 x=-3$$