Solve 16x^2+160x-48000=0

$$16 { x }^{ 2 } +160x-48000=0$$

Solve for x

$$x=-60$$

$$x=50$$

Steps Using Factoring By Grouping

Divide both sides by 16.

$$x^{2}+10x-3000=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-3000. To find a and b, set up a system to be solved.

$$a+b=10 ab=1\left(-3000\right)=-3000$$

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 -3000.

$$-1,3000 -2,1500 -3,1000 -4,750 -5,600 -6,500 -8,375 -10,300 -12,250 -15,200 -20,150 -24,125 -25,120 -30,100 -40,75 -50,60$$

Calculate the sum for each pair.

$$-1+3000=2999 -2+1500=1498 -3+1000=997 -4+750=746 -5+600=595 -6+500=494 -8+375=367 -10+300=290 -12+250=238 -15+200=185 -20+150=130 -24+125=101 -25+120=95 -30+100=70 -40+75=35 -50+60=10$$

The solution is the pair that gives sum 10.

$$a=-50 b=60$$

Rewrite x^{2}+10x-3000 as \left(x^{2}-50x\right)+\left(60x-3000\right).

$$\left(x^{2}-50x\right)+\left(60x-3000\right)$$

Factor out x in the first and 60 in the second group.

$$x\left(x-50\right)+60\left(x-50\right)$$

Factor out common term x-50 by using distributive property.

$$\left(x-50\right)\left(x+60\right)$$

To find equation solutions, solve x-50=0 and x+60=0.

$$x=50 x=-60$$

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.

$$16x^{2}+160x-48000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, 160 for b, and -48000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-160±\sqrt{160^{2}-4\times 16\left(-48000\right)}}{2\times 16}$$

Square 160.

$$x=\frac{-160±\sqrt{25600-4\times 16\left(-48000\right)}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-160±\sqrt{25600-64\left(-48000\right)}}{2\times 16}$$

Multiply -64 times -48000.

$$x=\frac{-160±\sqrt{25600+3072000}}{2\times 16}$$

Add 25600 to 3072000.

$$x=\frac{-160±\sqrt{3097600}}{2\times 16}$$

Take the square root of 3097600.

$$x=\frac{-160±1760}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{-160±1760}{32}$$

Now solve the equation x=\frac{-160±1760}{32} when ± is plus. Add -160 to 1760.

$$x=\frac{1600}{32}$$

Divide 1600 by 32.

$$x=50$$

Now solve the equation x=\frac{-160±1760}{32} when ± is minus. Subtract 1760 from -160.

$$x=-\frac{1920}{32}$$

Divide -1920 by 32.

$$x=-60$$

The equation is now solved.

$$x=50 x=-60$$

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.

$$16x^{2}+160x-48000=0$$

Add 48000 to both sides of the equation.

$$16x^{2}+160x-48000-\left(-48000\right)=-\left(-48000\right)$$

Subtracting -48000 from itself leaves 0.

$$16x^{2}+160x=-\left(-48000\right)$$

Subtract -48000 from 0.

$$16x^{2}+160x=48000$$

Divide both sides by 16.

$$\frac{16x^{2}+160x}{16}=\frac{48000}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$x^{2}+\frac{160}{16}x=\frac{48000}{16}$$

Divide 160 by 16.

$$x^{2}+10x=\frac{48000}{16}$$

Divide 48000 by 16.

$$x^{2}+10x=3000$$

Divide 10, the coefficient of the x term, by 2 to get 5. Then add the square of 5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+10x+5^{2}=3000+5^{2}$$

Square 5.

$$x^{2}+10x+25=3000+25$$

Add 3000 to 25.

$$x^{2}+10x+25=3025$$

Factor x^{2}+10x+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}.

$$\left(x+5\right)^{2}=3025$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{3025}$$

Simplify.

$$x+5=55 x+5=-55$$

Subtract 5 from both sides of the equation.

$$x=50 x=-60$$