Solve 3x^2-360x/120-x=498

$$\frac { 3 x ^ { 2 } - 360 x } { 120 - x } = 498$$

Solve for x

$$x=-166$$

Steps Using Factoring By Grouping

Variable x cannot be equal to 120 since division by zero is not defined. Multiply both sides of the equation by -x+120.

$$3x^{2}-360x=498\left(-x+120\right)$$

Use the distributive property to multiply 498 by -x+120.

$$3x^{2}-360x=-498x+59760$$

Add 498x to both sides.

$$3x^{2}-360x+498x=59760$$

Combine -360x and 498x to get 138x.

$$3x^{2}+138x=59760$$

Subtract 59760 from both sides.

$$3x^{2}+138x-59760=0$$

Divide both sides by 3.

$$x^{2}+46x-19920=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-19920. To find a and b, set up a system to be solved.

$$a+b=46 ab=1\left(-19920\right)=-19920$$

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 -19920.

$$-1,19920 -2,9960 -3,6640 -4,4980 -5,3984 -6,3320 -8,2490 -10,1992 -12,1660 -15,1328 -16,1245 -20,996 -24,830 -30,664 -40,498 -48,415 -60,332 -80,249 -83,240 -120,166$$

Calculate the sum for each pair.

$$-1+19920=19919 -2+9960=9958 -3+6640=6637 -4+4980=4976 -5+3984=3979 -6+3320=3314 -8+2490=2482 -10+1992=1982 -12+1660=1648 -15+1328=1313 -16+1245=1229 -20+996=976 -24+830=806 -30+664=634 -40+498=458 -48+415=367 -60+332=272 -80+249=169 -83+240=157 -120+166=46$$

The solution is the pair that gives sum 46.

$$a=-120 b=166$$

Rewrite x^{2}+46x-19920 as \left(x^{2}-120x\right)+\left(166x-19920\right).

$$\left(x^{2}-120x\right)+\left(166x-19920\right)$$

Factor out x in the first and 166 in the second group.

$$x\left(x-120\right)+166\left(x-120\right)$$

Factor out common term x-120 by using distributive property.

$$\left(x-120\right)\left(x+166\right)$$

To find equation solutions, solve x-120=0 and x+166=0.

$$x=120 x=-166$$

Variable x cannot be equal to 120.

$$x=-166$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 120 since division by zero is not defined. Multiply both sides of the equation by -x+120.

$$3x^{2}-360x=498\left(-x+120\right)$$

Use the distributive property to multiply 498 by -x+120.

$$3x^{2}-360x=-498x+59760$$

Add 498x to both sides.

$$3x^{2}-360x+498x=59760$$

Combine -360x and 498x to get 138x.

$$3x^{2}+138x=59760$$

Subtract 59760 from both sides.

$$3x^{2}+138x-59760=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 138 for b, and -59760 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-138±\sqrt{138^{2}-4\times 3\left(-59760\right)}}{2\times 3}$$

Square 138.

$$x=\frac{-138±\sqrt{19044-4\times 3\left(-59760\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-138±\sqrt{19044-12\left(-59760\right)}}{2\times 3}$$

Multiply -12 times -59760.

$$x=\frac{-138±\sqrt{19044+717120}}{2\times 3}$$

Add 19044 to 717120.

$$x=\frac{-138±\sqrt{736164}}{2\times 3}$$

Take the square root of 736164.

$$x=\frac{-138±858}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-138±858}{6}$$

Now solve the equation x=\frac{-138±858}{6} when ± is plus. Add -138 to 858.

$$x=\frac{720}{6}$$

Divide 720 by 6.

$$x=120$$

Now solve the equation x=\frac{-138±858}{6} when ± is minus. Subtract 858 from -138.

$$x=-\frac{996}{6}$$

Divide -996 by 6.

$$x=-166$$

The equation is now solved.

$$x=120 x=-166$$

Variable x cannot be equal to 120.

$$x=-166$$

Steps for Completing the Square

Variable x cannot be equal to 120 since division by zero is not defined. Multiply both sides of the equation by -x+120.

$$3x^{2}-360x=498\left(-x+120\right)$$

Use the distributive property to multiply 498 by -x+120.

$$3x^{2}-360x=-498x+59760$$

Add 498x to both sides.

$$3x^{2}-360x+498x=59760$$

Combine -360x and 498x to get 138x.

$$3x^{2}+138x=59760$$

Divide both sides by 3.

$$\frac{3x^{2}+138x}{3}=\frac{59760}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\frac{138}{3}x=\frac{59760}{3}$$

Divide 138 by 3.

$$x^{2}+46x=\frac{59760}{3}$$

Divide 59760 by 3.

$$x^{2}+46x=19920$$

Divide 46, the coefficient of the x term, by 2 to get 23. Then add the square of 23 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+46x+23^{2}=19920+23^{2}$$

Square 23.

$$x^{2}+46x+529=19920+529$$

Add 19920 to 529.

$$x^{2}+46x+529=20449$$

Factor x^{2}+46x+529. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+23\right)^{2}=20449$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+23\right)^{2}}=\sqrt{20449}$$

Simplify.

$$x+23=143 x+23=-143$$

Subtract 23 from both sides of the equation.

$$x=120 x=-166$$

Variable x cannot be equal to 120.

$$x=-166$$