Solve 5x^2+125x-18750=0

$$5 { x }^{ 2 } +125x-18750=0$$

Solve for x

$$x=-75$$

$$x=50$$

Steps Using Factoring By Grouping

Divide both sides by 5.

$$x^{2}+25x-3750=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-3750. To find a and b, set up a system to be solved.

$$a+b=25 ab=1\left(-3750\right)=-3750$$

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 -3750.

$$-1,3750 -2,1875 -3,1250 -5,750 -6,625 -10,375 -15,250 -25,150 -30,125 -50,75$$

Calculate the sum for each pair.

$$-1+3750=3749 -2+1875=1873 -3+1250=1247 -5+750=745 -6+625=619 -10+375=365 -15+250=235 -25+150=125 -30+125=95 -50+75=25$$

The solution is the pair that gives sum 25.

$$a=-50 b=75$$

Rewrite x^{2}+25x-3750 as \left(x^{2}-50x\right)+\left(75x-3750\right).

$$\left(x^{2}-50x\right)+\left(75x-3750\right)$$

Factor out x in the first and 75 in the second group.

$$x\left(x-50\right)+75\left(x-50\right)$$

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

$$\left(x-50\right)\left(x+75\right)$$

To find equation solutions, solve x-50=0 and x+75=0.

$$x=50 x=-75$$

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.

$$5x^{2}+125x-18750=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 125 for b, and -18750 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-125±\sqrt{125^{2}-4\times 5\left(-18750\right)}}{2\times 5}$$

Square 125.

$$x=\frac{-125±\sqrt{15625-4\times 5\left(-18750\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-125±\sqrt{15625-20\left(-18750\right)}}{2\times 5}$$

Multiply -20 times -18750.

$$x=\frac{-125±\sqrt{15625+375000}}{2\times 5}$$

Add 15625 to 375000.

$$x=\frac{-125±\sqrt{390625}}{2\times 5}$$

Take the square root of 390625.

$$x=\frac{-125±625}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-125±625}{10}$$

Now solve the equation x=\frac{-125±625}{10} when ± is plus. Add -125 to 625.

$$x=\frac{500}{10}$$

Divide 500 by 10.

$$x=50$$

Now solve the equation x=\frac{-125±625}{10} when ± is minus. Subtract 625 from -125.

$$x=-\frac{750}{10}$$

Divide -750 by 10.

$$x=-75$$

The equation is now solved.

$$x=50 x=-75$$

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.

$$5x^{2}+125x-18750=0$$

Add 18750 to both sides of the equation.

$$5x^{2}+125x-18750-\left(-18750\right)=-\left(-18750\right)$$

Subtracting -18750 from itself leaves 0.

$$5x^{2}+125x=-\left(-18750\right)$$

Subtract -18750 from 0.

$$5x^{2}+125x=18750$$

Divide both sides by 5.

$$\frac{5x^{2}+125x}{5}=\frac{18750}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{125}{5}x=\frac{18750}{5}$$

Divide 125 by 5.

$$x^{2}+25x=\frac{18750}{5}$$

Divide 18750 by 5.

$$x^{2}+25x=3750$$

Divide 25, the coefficient of the x term, by 2 to get \frac{25}{2}. Then add the square of \frac{25}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+25x+\left(\frac{25}{2}\right)^{2}=3750+\left(\frac{25}{2}\right)^{2}$$

Square \frac{25}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+25x+\frac{625}{4}=3750+\frac{625}{4}$$

Add 3750 to \frac{625}{4}.

$$x^{2}+25x+\frac{625}{4}=\frac{15625}{4}$$

Factor x^{2}+25x+\frac{625}{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}.

$$\left(x+\frac{25}{2}\right)^{2}=\frac{15625}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{25}{2}\right)^{2}}=\sqrt{\frac{15625}{4}}$$

Simplify.

$$x+\frac{25}{2}=\frac{125}{2} x+\frac{25}{2}=-\frac{125}{2}$$

Subtract \frac{25}{2} from both sides of the equation.

$$x=50 x=-75$$