Solve x/3+12/x=5(x-1)/2

$$\frac { x } { 3 } + \frac { 12 } { x } = \frac { 5 ( x - 1 ) } { 2 }$$

Solve for x

$$x = -\frac{24}{13} = -1\frac{11}{13} \approx -1.846153846$$

$$x=3$$

Steps Using Factoring By Grouping

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 6x, the least common multiple of 3,x,2.

$$2xx+6\times 12=3x\times 5\left(x-1\right)$$

Multiply x and x to get x^{2}.

$$2x^{2}+6\times 12=3x\times 5\left(x-1\right)$$

Multiply 6 and 12 to get 72.

$$2x^{2}+72=3x\times 5\left(x-1\right)$$

Multiply 3 and 5 to get 15.

$$2x^{2}+72=15x\left(x-1\right)$$

Use the distributive property to multiply 15x by x-1.

$$2x^{2}+72=15x^{2}-15x$$

Subtract 15x^{2} from both sides.

$$2x^{2}+72-15x^{2}=-15x$$

Combine 2x^{2} and -15x^{2} to get -13x^{2}.

$$-13x^{2}+72=-15x$$

Add 15x to both sides.

$$-13x^{2}+72+15x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-13x^{2}+15x+72=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -13x^{2}+ax+bx+72. To find a and b, set up a system to be solved.

$$a+b=15 ab=-13\times 72=-936$$

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 -936.

$$-1,936 -2,468 -3,312 -4,234 -6,156 -8,117 -9,104 -12,78 -13,72 -18,52 -24,39 -26,36$$

Calculate the sum for each pair.

$$-1+936=935 -2+468=466 -3+312=309 -4+234=230 -6+156=150 -8+117=109 -9+104=95 -12+78=66 -13+72=59 -18+52=34 -24+39=15 -26+36=10$$

The solution is the pair that gives sum 15.

$$a=39 b=-24$$

Rewrite -13x^{2}+15x+72 as \left(-13x^{2}+39x\right)+\left(-24x+72\right).

$$\left(-13x^{2}+39x\right)+\left(-24x+72\right)$$

Factor out 13x in the first and 24 in the second group.

$$13x\left(-x+3\right)+24\left(-x+3\right)$$

Factor out common term -x+3 by using distributive property.

$$\left(-x+3\right)\left(13x+24\right)$$

To find equation solutions, solve -x+3=0 and 13x+24=0.

$$x=3 x=-\frac{24}{13}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 6x, the least common multiple of 3,x,2.

$$2xx+6\times 12=3x\times 5\left(x-1\right)$$

Multiply x and x to get x^{2}.

$$2x^{2}+6\times 12=3x\times 5\left(x-1\right)$$

Multiply 6 and 12 to get 72.

$$2x^{2}+72=3x\times 5\left(x-1\right)$$

Multiply 3 and 5 to get 15.

$$2x^{2}+72=15x\left(x-1\right)$$

Use the distributive property to multiply 15x by x-1.

$$2x^{2}+72=15x^{2}-15x$$

Subtract 15x^{2} from both sides.

$$2x^{2}+72-15x^{2}=-15x$$

Combine 2x^{2} and -15x^{2} to get -13x^{2}.

$$-13x^{2}+72=-15x$$

Add 15x to both sides.

$$-13x^{2}+72+15x=0$$

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.

$$-13x^{2}+15x+72=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -13 for a, 15 for b, and 72 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15±\sqrt{15^{2}-4\left(-13\right)\times 72}}{2\left(-13\right)}$$

Square 15.

$$x=\frac{-15±\sqrt{225-4\left(-13\right)\times 72}}{2\left(-13\right)}$$

Multiply -4 times -13.

$$x=\frac{-15±\sqrt{225+52\times 72}}{2\left(-13\right)}$$

Multiply 52 times 72.

$$x=\frac{-15±\sqrt{225+3744}}{2\left(-13\right)}$$

Add 225 to 3744.

$$x=\frac{-15±\sqrt{3969}}{2\left(-13\right)}$$

Take the square root of 3969.

$$x=\frac{-15±63}{2\left(-13\right)}$$

Multiply 2 times -13.

$$x=\frac{-15±63}{-26}$$

Now solve the equation x=\frac{-15±63}{-26} when ± is plus. Add -15 to 63.

$$x=\frac{48}{-26}$$

Reduce the fraction \frac{48}{-26} to lowest terms by extracting and canceling out 2.

$$x=-\frac{24}{13}$$

Now solve the equation x=\frac{-15±63}{-26} when ± is minus. Subtract 63 from -15.

$$x=-\frac{78}{-26}$$

Divide -78 by -26.

$$x=3$$

The equation is now solved.

$$x=-\frac{24}{13} x=3$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 6x, the least common multiple of 3,x,2.

$$2xx+6\times 12=3x\times 5\left(x-1\right)$$

Multiply x and x to get x^{2}.

$$2x^{2}+6\times 12=3x\times 5\left(x-1\right)$$

Multiply 6 and 12 to get 72.

$$2x^{2}+72=3x\times 5\left(x-1\right)$$

Multiply 3 and 5 to get 15.

$$2x^{2}+72=15x\left(x-1\right)$$

Use the distributive property to multiply 15x by x-1.

$$2x^{2}+72=15x^{2}-15x$$

Subtract 15x^{2} from both sides.

$$2x^{2}+72-15x^{2}=-15x$$

Combine 2x^{2} and -15x^{2} to get -13x^{2}.

$$-13x^{2}+72=-15x$$

Add 15x to both sides.

$$-13x^{2}+72+15x=0$$

Subtract 72 from both sides. Anything subtracted from zero gives its negation.

$$-13x^{2}+15x=-72$$

Divide both sides by -13.

$$\frac{-13x^{2}+15x}{-13}=-\frac{72}{-13}$$

Dividing by -13 undoes the multiplication by -13.

$$x^{2}+\frac{15}{-13}x=-\frac{72}{-13}$$

Divide 15 by -13.

$$x^{2}-\frac{15}{13}x=-\frac{72}{-13}$$

Divide -72 by -13.

$$x^{2}-\frac{15}{13}x=\frac{72}{13}$$

Divide -\frac{15}{13}, the coefficient of the x term, by 2 to get -\frac{15}{26}. Then add the square of -\frac{15}{26} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{15}{13}x+\left(-\frac{15}{26}\right)^{2}=\frac{72}{13}+\left(-\frac{15}{26}\right)^{2}$$

Square -\frac{15}{26} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{15}{13}x+\frac{225}{676}=\frac{72}{13}+\frac{225}{676}$$

Add \frac{72}{13} to \frac{225}{676} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{15}{13}x+\frac{225}{676}=\frac{3969}{676}$$

Factor x^{2}-\frac{15}{13}x+\frac{225}{676}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{15}{26}\right)^{2}=\frac{3969}{676}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{26}\right)^{2}}=\sqrt{\frac{3969}{676}}$$

Simplify.

$$x-\frac{15}{26}=\frac{63}{26} x-\frac{15}{26}=-\frac{63}{26}$$

Add \frac{15}{26} to both sides of the equation.

$$x=3 x=-\frac{24}{13}$$