Solve x^2+201x-5400=0

$${ x }^{ 2 } +201x-5400=0$$

Solve for x

$$x=-225$$

$$x=24$$

Steps Using Factoring

To solve the equation, factor x^{2}+201x-5400 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.

$$a+b=201 ab=-5400$$

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 -5400.

$$-1,5400 -2,2700 -3,1800 -4,1350 -5,1080 -6,900 -8,675 -9,600 -10,540 -12,450 -15,360 -18,300 -20,270 -24,225 -25,216 -27,200 -30,180 -36,150 -40,135 -45,120 -50,108 -54,100 -60,90 -72,75$$

Calculate the sum for each pair.

$$-1+5400=5399 -2+2700=2698 -3+1800=1797 -4+1350=1346 -5+1080=1075 -6+900=894 -8+675=667 -9+600=591 -10+540=530 -12+450=438 -15+360=345 -18+300=282 -20+270=250 -24+225=201 -25+216=191 -27+200=173 -30+180=150 -36+150=114 -40+135=95 -45+120=75 -50+108=58 -54+100=46 -60+90=30 -72+75=3$$

The solution is the pair that gives sum 201.

$$a=-24 b=225$$

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

$$\left(x-24\right)\left(x+225\right)$$

To find equation solutions, solve x-24=0 and x+225=0.

$$x=24 x=-225$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-5400. To find a and b, set up a system to be solved.

$$a+b=201 ab=1\left(-5400\right)=-5400$$

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 -5400.

$$-1,5400 -2,2700 -3,1800 -4,1350 -5,1080 -6,900 -8,675 -9,600 -10,540 -12,450 -15,360 -18,300 -20,270 -24,225 -25,216 -27,200 -30,180 -36,150 -40,135 -45,120 -50,108 -54,100 -60,90 -72,75$$

Calculate the sum for each pair.

$$-1+5400=5399 -2+2700=2698 -3+1800=1797 -4+1350=1346 -5+1080=1075 -6+900=894 -8+675=667 -9+600=591 -10+540=530 -12+450=438 -15+360=345 -18+300=282 -20+270=250 -24+225=201 -25+216=191 -27+200=173 -30+180=150 -36+150=114 -40+135=95 -45+120=75 -50+108=58 -54+100=46 -60+90=30 -72+75=3$$

The solution is the pair that gives sum 201.

$$a=-24 b=225$$

Rewrite x^{2}+201x-5400 as \left(x^{2}-24x\right)+\left(225x-5400\right).

$$\left(x^{2}-24x\right)+\left(225x-5400\right)$$

Factor out x in the first and 225 in the second group.

$$x\left(x-24\right)+225\left(x-24\right)$$

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

$$\left(x-24\right)\left(x+225\right)$$

To find equation solutions, solve x-24=0 and x+225=0.

$$x=24 x=-225$$

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.

$$x^{2}+201x-5400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 201 for b, and -5400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-201±\sqrt{201^{2}-4\left(-5400\right)}}{2}$$

Square 201.

$$x=\frac{-201±\sqrt{40401-4\left(-5400\right)}}{2}$$

Multiply -4 times -5400.

$$x=\frac{-201±\sqrt{40401+21600}}{2}$$

Add 40401 to 21600.

$$x=\frac{-201±\sqrt{62001}}{2}$$

Take the square root of 62001.

$$x=\frac{-201±249}{2}$$

Now solve the equation x=\frac{-201±249}{2} when ± is plus. Add -201 to 249.

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

Divide 48 by 2.

$$x=24$$

Now solve the equation x=\frac{-201±249}{2} when ± is minus. Subtract 249 from -201.

$$x=-\frac{450}{2}$$

Divide -450 by 2.

$$x=-225$$

The equation is now solved.

$$x=24 x=-225$$

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.

$$x^{2}+201x-5400=0$$

Add 5400 to both sides of the equation.

$$x^{2}+201x-5400-\left(-5400\right)=-\left(-5400\right)$$

Subtracting -5400 from itself leaves 0.

$$x^{2}+201x=-\left(-5400\right)$$

Subtract -5400 from 0.

$$x^{2}+201x=5400$$

Divide 201, the coefficient of the x term, by 2 to get \frac{201}{2}. Then add the square of \frac{201}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+201x+\left(\frac{201}{2}\right)^{2}=5400+\left(\frac{201}{2}\right)^{2}$$

Square \frac{201}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+201x+\frac{40401}{4}=5400+\frac{40401}{4}$$

Add 5400 to \frac{40401}{4}.

$$x^{2}+201x+\frac{40401}{4}=\frac{62001}{4}$$

Factor x^{2}+201x+\frac{40401}{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}.

$$\left(x+\frac{201}{2}\right)^{2}=\frac{62001}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{201}{2}\right)^{2}}=\sqrt{\frac{62001}{4}}$$

Simplify.

$$x+\frac{201}{2}=\frac{249}{2} x+\frac{201}{2}=-\frac{249}{2}$$

Subtract \frac{201}{2} from both sides of the equation.

$$x=24 x=-225$$