Solve 96(x-4)+96(x+4)=10(x^2-16)

$$96 ( x - 4 ) + 96 ( x + 4 ) = 10 ( x ^ { 2 } - 16 )$$

Solve for x

$$x=20$$

$$x=-\frac{4}{5}=-0.8$$

Steps Using Factoring By Grouping

Use the distributive property to multiply 96 by x-4.

$$96x-384+96\left(x+4\right)=10\left(x^{2}-16\right)$$

Use the distributive property to multiply 96 by x+4.

$$96x-384+96x+384=10\left(x^{2}-16\right)$$

Combine 96x and 96x to get 192x.

$$192x-384+384=10\left(x^{2}-16\right)$$

Add -384 and 384 to get 0.

$$192x=10\left(x^{2}-16\right)$$

Use the distributive property to multiply 10 by x^{2}-16.

$$192x=10x^{2}-160$$

Subtract 10x^{2} from both sides.

$$192x-10x^{2}=-160$$

Add 160 to both sides.

$$192x-10x^{2}+160=0$$

Divide both sides by 2.

$$96x-5x^{2}+80=0$$

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

$$-5x^{2}+96x+80=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -5x^{2}+ax+bx+80. To find a and b, set up a system to be solved.

$$a+b=96 ab=-5\times 80=-400$$

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 -400.

$$-1,400 -2,200 -4,100 -5,80 -8,50 -10,40 -16,25 -20,20$$

Calculate the sum for each pair.

$$-1+400=399 -2+200=198 -4+100=96 -5+80=75 -8+50=42 -10+40=30 -16+25=9 -20+20=0$$

The solution is the pair that gives sum 96.

$$a=100 b=-4$$

Rewrite -5x^{2}+96x+80 as \left(-5x^{2}+100x\right)+\left(-4x+80\right).

$$\left(-5x^{2}+100x\right)+\left(-4x+80\right)$$

Factor out 5x in the first and 4 in the second group.

$$5x\left(-x+20\right)+4\left(-x+20\right)$$

Factor out common term -x+20 by using distributive property.

$$\left(-x+20\right)\left(5x+4\right)$$

To find equation solutions, solve -x+20=0 and 5x+4=0.

$$x=20 x=-\frac{4}{5}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 96 by x-4.

$$96x-384+96\left(x+4\right)=10\left(x^{2}-16\right)$$

Use the distributive property to multiply 96 by x+4.

$$96x-384+96x+384=10\left(x^{2}-16\right)$$

Combine 96x and 96x to get 192x.

$$192x-384+384=10\left(x^{2}-16\right)$$

Add -384 and 384 to get 0.

$$192x=10\left(x^{2}-16\right)$$

Use the distributive property to multiply 10 by x^{2}-16.

$$192x=10x^{2}-160$$

Subtract 10x^{2} from both sides.

$$192x-10x^{2}=-160$$

Add 160 to both sides.

$$192x-10x^{2}+160=0$$

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.

$$-10x^{2}+192x+160=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 192 for b, and 160 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-192±\sqrt{192^{2}-4\left(-10\right)\times 160}}{2\left(-10\right)}$$

Square 192.

$$x=\frac{-192±\sqrt{36864-4\left(-10\right)\times 160}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-192±\sqrt{36864+40\times 160}}{2\left(-10\right)}$$

Multiply 40 times 160.

$$x=\frac{-192±\sqrt{36864+6400}}{2\left(-10\right)}$$

Add 36864 to 6400.

$$x=\frac{-192±\sqrt{43264}}{2\left(-10\right)}$$

Take the square root of 43264.

$$x=\frac{-192±208}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-192±208}{-20}$$

Now solve the equation x=\frac{-192±208}{-20} when ± is plus. Add -192 to 208.

$$x=\frac{16}{-20}$$

Reduce the fraction \frac{16}{-20} to lowest terms by extracting and canceling out 4.

$$x=-\frac{4}{5}$$

Now solve the equation x=\frac{-192±208}{-20} when ± is minus. Subtract 208 from -192.

$$x=-\frac{400}{-20}$$

Divide -400 by -20.

$$x=20$$

The equation is now solved.

$$x=-\frac{4}{5} x=20$$

Steps for Completing the Square

Use the distributive property to multiply 96 by x-4.

$$96x-384+96\left(x+4\right)=10\left(x^{2}-16\right)$$

Use the distributive property to multiply 96 by x+4.

$$96x-384+96x+384=10\left(x^{2}-16\right)$$

Combine 96x and 96x to get 192x.

$$192x-384+384=10\left(x^{2}-16\right)$$

Add -384 and 384 to get 0.

$$192x=10\left(x^{2}-16\right)$$

Use the distributive property to multiply 10 by x^{2}-16.

$$192x=10x^{2}-160$$

Subtract 10x^{2} from both sides.

$$192x-10x^{2}=-160$$

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.

$$-10x^{2}+192x=-160$$

Divide both sides by -10.

$$\frac{-10x^{2}+192x}{-10}=-\frac{160}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{192}{-10}x=-\frac{160}{-10}$$

Reduce the fraction \frac{192}{-10} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{96}{5}x=-\frac{160}{-10}$$

Divide -160 by -10.

$$x^{2}-\frac{96}{5}x=16$$

Divide -\frac{96}{5}, the coefficient of the x term, by 2 to get -\frac{48}{5}. Then add the square of -\frac{48}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{96}{5}x+\left(-\frac{48}{5}\right)^{2}=16+\left(-\frac{48}{5}\right)^{2}$$

Square -\frac{48}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{96}{5}x+\frac{2304}{25}=16+\frac{2304}{25}$$

Add 16 to \frac{2304}{25}.

$$x^{2}-\frac{96}{5}x+\frac{2304}{25}=\frac{2704}{25}$$

Factor x^{2}-\frac{96}{5}x+\frac{2304}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{48}{5}\right)^{2}=\frac{2704}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{48}{5}\right)^{2}}=\sqrt{\frac{2704}{25}}$$

Simplify.

$$x-\frac{48}{5}=\frac{52}{5} x-\frac{48}{5}=-\frac{52}{5}$$

Add \frac{48}{5} to both sides of the equation.

$$x=20 x=-\frac{4}{5}$$