Solve x^2-16x=12*16

$${ x }^{ 2 } -16x=12 \times 16$$

Solve for x

$$x=-8$$

$$x=24$$

Steps Using Factoring

Multiply 12 and 16 to get 192.

$$x^{2}-16x=192$$

Subtract 192 from both sides.

$$x^{2}-16x-192=0$$

To solve the equation, factor x^{2}-16x-192 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.

$$a+b=-16 ab=-192$$

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 -192.

$$1,-192 2,-96 3,-64 4,-48 6,-32 8,-24 12,-16$$

Calculate the sum for each pair.

$$1-192=-191 2-96=-94 3-64=-61 4-48=-44 6-32=-26 8-24=-16 12-16=-4$$

The solution is the pair that gives sum -16.

$$a=-24 b=8$$

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

$$\left(x-24\right)\left(x+8\right)$$

To find equation solutions, solve x-24=0 and x+8=0.

$$x=24 x=-8$$

Steps Using Factoring By Grouping

Multiply 12 and 16 to get 192.

$$x^{2}-16x=192$$

Subtract 192 from both sides.

$$x^{2}-16x-192=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-192. To find a and b, set up a system to be solved.

$$a+b=-16 ab=1\left(-192\right)=-192$$

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 -192.

$$1,-192 2,-96 3,-64 4,-48 6,-32 8,-24 12,-16$$

Calculate the sum for each pair.

$$1-192=-191 2-96=-94 3-64=-61 4-48=-44 6-32=-26 8-24=-16 12-16=-4$$

The solution is the pair that gives sum -16.

$$a=-24 b=8$$

Rewrite x^{2}-16x-192 as \left(x^{2}-24x\right)+\left(8x-192\right).

$$\left(x^{2}-24x\right)+\left(8x-192\right)$$

Factor out x in the first and 8 in the second group.

$$x\left(x-24\right)+8\left(x-24\right)$$

Factor out common term x-24 by using distributive property.

$$\left(x-24\right)\left(x+8\right)$$

To find equation solutions, solve x-24=0 and x+8=0.

$$x=24 x=-8$$

Steps Using the Quadratic Formula

Multiply 12 and 16 to get 192.

$$x^{2}-16x=192$$

Subtract 192 from both sides.

$$x^{2}-16x-192=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -16 for b, and -192 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\left(-192\right)}}{2}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\left(-192\right)}}{2}$$

Multiply -4 times -192.

$$x=\frac{-\left(-16\right)±\sqrt{256+768}}{2}$$

Add 256 to 768.

$$x=\frac{-\left(-16\right)±\sqrt{1024}}{2}$$

Take the square root of 1024.

$$x=\frac{-\left(-16\right)±32}{2}$$

The opposite of -16 is 16.

$$x=\frac{16±32}{2}$$

Now solve the equation x=\frac{16±32}{2} when ± is plus. Add 16 to 32.

$$x=\frac{48}{2}$$

Divide 48 by 2.

$$x=24$$

Now solve the equation x=\frac{16±32}{2} when ± is minus. Subtract 32 from 16.

$$x=-\frac{16}{2}$$

Divide -16 by 2.

$$x=-8$$

The equation is now solved.

$$x=24 x=-8$$

Steps for Completing the Square

Multiply 12 and 16 to get 192.

$$x^{2}-16x=192$$

Divide -16, the coefficient of the x term, by 2 to get -8. Then add the square of -8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-16x+\left(-8\right)^{2}=192+\left(-8\right)^{2}$$

Square -8.

$$x^{2}-16x+64=192+64$$

Add 192 to 64.

$$x^{2}-16x+64=256$$

Factor x^{2}-16x+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-8\right)^{2}=256$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-8\right)^{2}}=\sqrt{256}$$

Simplify.

$$x-8=16 x-8=-16$$

Add 8 to both sides of the equation.

$$x=24 x=-8$$