Solve 25x^2+25x-33300=0

$$25 { x }^{ 2 } +25x-33300=0$$

Solve for x

$$x=-37$$

$$x=36$$

Steps Using Factoring By Grouping

Divide both sides by 25.

$$x^{2}+x-1332=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-1332. To find a and b, set up a system to be solved.

$$a+b=1 ab=1\left(-1332\right)=-1332$$

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 -1332.

$$-1,1332 -2,666 -3,444 -4,333 -6,222 -9,148 -12,111 -18,74 -36,37$$

Calculate the sum for each pair.

$$-1+1332=1331 -2+666=664 -3+444=441 -4+333=329 -6+222=216 -9+148=139 -12+111=99 -18+74=56 -36+37=1$$

The solution is the pair that gives sum 1.

$$a=-36 b=37$$

Rewrite x^{2}+x-1332 as \left(x^{2}-36x\right)+\left(37x-1332\right).

$$\left(x^{2}-36x\right)+\left(37x-1332\right)$$

Factor out x in the first and 37 in the second group.

$$x\left(x-36\right)+37\left(x-36\right)$$

Factor out common term x-36 by using distributive property.

$$\left(x-36\right)\left(x+37\right)$$

To find equation solutions, solve x-36=0 and x+37=0.

$$x=36 x=-37$$

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.

$$25x^{2}+25x-33300=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 25 for a, 25 for b, and -33300 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-25±\sqrt{25^{2}-4\times 25\left(-33300\right)}}{2\times 25}$$

Square 25.

$$x=\frac{-25±\sqrt{625-4\times 25\left(-33300\right)}}{2\times 25}$$

Multiply -4 times 25.

$$x=\frac{-25±\sqrt{625-100\left(-33300\right)}}{2\times 25}$$

Multiply -100 times -33300.

$$x=\frac{-25±\sqrt{625+3330000}}{2\times 25}$$

Add 625 to 3330000.

$$x=\frac{-25±\sqrt{3330625}}{2\times 25}$$

Take the square root of 3330625.

$$x=\frac{-25±1825}{2\times 25}$$

Multiply 2 times 25.

$$x=\frac{-25±1825}{50}$$

Now solve the equation x=\frac{-25±1825}{50} when ± is plus. Add -25 to 1825.

$$x=\frac{1800}{50}$$

Divide 1800 by 50.

$$x=36$$

Now solve the equation x=\frac{-25±1825}{50} when ± is minus. Subtract 1825 from -25.

$$x=-\frac{1850}{50}$$

Divide -1850 by 50.

$$x=-37$$

The equation is now solved.

$$x=36 x=-37$$

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.

$$25x^{2}+25x-33300=0$$

Add 33300 to both sides of the equation.

$$25x^{2}+25x-33300-\left(-33300\right)=-\left(-33300\right)$$

Subtracting -33300 from itself leaves 0.

$$25x^{2}+25x=-\left(-33300\right)$$

Subtract -33300 from 0.

$$25x^{2}+25x=33300$$

Divide both sides by 25.

$$\frac{25x^{2}+25x}{25}=\frac{33300}{25}$$

Dividing by 25 undoes the multiplication by 25.

$$x^{2}+\frac{25}{25}x=\frac{33300}{25}$$

Divide 25 by 25.

$$x^{2}+x=\frac{33300}{25}$$

Divide 33300 by 25.

$$x^{2}+x=1332$$

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=1332+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=1332+\frac{1}{4}$$

Add 1332 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{5329}{4}$$

Factor x^{2}+x+\frac{1}{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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{5329}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{5329}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{73}{2} x+\frac{1}{2}=-\frac{73}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=36 x=-37$$