Solve 2t-6/16+15-2t/12-5t=4t-15/18

$$\frac{ 2t-6 }{ 16 } + \frac{ 15-2t }{ 12-5t } = \frac{ 4t-15 }{ 18 }$$

Solve for t

$$t=\frac{393+3\sqrt{5799}i}{70}\approx 5.614285714+3.263621377i$$

$$t=\frac{-3\sqrt{5799}i+393}{70}\approx 5.614285714-3.263621377i$$

Steps Using the Quadratic Formula

Variable t cannot be equal to \frac{12}{5} since division by zero is not defined. Multiply both sides of the equation by 144\left(5t-12\right), the least common multiple of 16,12-5t,18.

$$\left(45t-108\right)\left(2t-6\right)-144\left(15-2t\right)=\left(40t-96\right)\left(4t-15\right)$$

Use the distributive property to multiply 45t-108 by 2t-6 and combine like terms.

$$90t^{2}-486t+648-144\left(15-2t\right)=\left(40t-96\right)\left(4t-15\right)$$

Use the distributive property to multiply -144 by 15-2t.

$$90t^{2}-486t+648-2160+288t=\left(40t-96\right)\left(4t-15\right)$$

Subtract 2160 from 648 to get -1512.

$$90t^{2}-486t-1512+288t=\left(40t-96\right)\left(4t-15\right)$$

Combine -486t and 288t to get -198t.

$$90t^{2}-198t-1512=\left(40t-96\right)\left(4t-15\right)$$

Use the distributive property to multiply 40t-96 by 4t-15 and combine like terms.

$$90t^{2}-198t-1512=160t^{2}-984t+1440$$

Subtract 160t^{2} from both sides.

$$90t^{2}-198t-1512-160t^{2}=-984t+1440$$

Combine 90t^{2} and -160t^{2} to get -70t^{2}.

$$-70t^{2}-198t-1512=-984t+1440$$

Add 984t to both sides.

$$-70t^{2}-198t-1512+984t=1440$$

Combine -198t and 984t to get 786t.

$$-70t^{2}+786t-1512=1440$$

Subtract 1440 from both sides.

$$-70t^{2}+786t-1512-1440=0$$

Subtract 1440 from -1512 to get -2952.

$$-70t^{2}+786t-2952=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -70 for a, 786 for b, and -2952 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-786±\sqrt{786^{2}-4\left(-70\right)\left(-2952\right)}}{2\left(-70\right)}$$

Square 786.

$$t=\frac{-786±\sqrt{617796-4\left(-70\right)\left(-2952\right)}}{2\left(-70\right)}$$

Multiply -4 times -70.

$$t=\frac{-786±\sqrt{617796+280\left(-2952\right)}}{2\left(-70\right)}$$

Multiply 280 times -2952.

$$t=\frac{-786±\sqrt{617796-826560}}{2\left(-70\right)}$$

Add 617796 to -826560.

$$t=\frac{-786±\sqrt{-208764}}{2\left(-70\right)}$$

Take the square root of -208764.

$$t=\frac{-786±6\sqrt{5799}i}{2\left(-70\right)}$$

Multiply 2 times -70.

$$t=\frac{-786±6\sqrt{5799}i}{-140}$$

Now solve the equation t=\frac{-786±6\sqrt{5799}i}{-140} when ± is plus. Add -786 to 6i\sqrt{5799}.

$$t=\frac{-786+6\sqrt{5799}i}{-140}$$

Divide -786+6i\sqrt{5799} by -140.

$$t=\frac{-3\sqrt{5799}i+393}{70}$$

Now solve the equation t=\frac{-786±6\sqrt{5799}i}{-140} when ± is minus. Subtract 6i\sqrt{5799} from -786.

$$t=\frac{-6\sqrt{5799}i-786}{-140}$$

Divide -786-6i\sqrt{5799} by -140.

$$t=\frac{393+3\sqrt{5799}i}{70}$$

The equation is now solved.

$$t=\frac{-3\sqrt{5799}i+393}{70} t=\frac{393+3\sqrt{5799}i}{70}$$

Steps for Completing the Square

Variable t cannot be equal to \frac{12}{5} since division by zero is not defined. Multiply both sides of the equation by 144\left(5t-12\right), the least common multiple of 16,12-5t,18.

$$\left(45t-108\right)\left(2t-6\right)-144\left(15-2t\right)=\left(40t-96\right)\left(4t-15\right)$$

Use the distributive property to multiply 45t-108 by 2t-6 and combine like terms.

$$90t^{2}-486t+648-144\left(15-2t\right)=\left(40t-96\right)\left(4t-15\right)$$

Use the distributive property to multiply -144 by 15-2t.

$$90t^{2}-486t+648-2160+288t=\left(40t-96\right)\left(4t-15\right)$$

Subtract 2160 from 648 to get -1512.

$$90t^{2}-486t-1512+288t=\left(40t-96\right)\left(4t-15\right)$$

Combine -486t and 288t to get -198t.

$$90t^{2}-198t-1512=\left(40t-96\right)\left(4t-15\right)$$

Use the distributive property to multiply 40t-96 by 4t-15 and combine like terms.

$$90t^{2}-198t-1512=160t^{2}-984t+1440$$

Subtract 160t^{2} from both sides.

$$90t^{2}-198t-1512-160t^{2}=-984t+1440$$

Combine 90t^{2} and -160t^{2} to get -70t^{2}.

$$-70t^{2}-198t-1512=-984t+1440$$

Add 984t to both sides.

$$-70t^{2}-198t-1512+984t=1440$$

Combine -198t and 984t to get 786t.

$$-70t^{2}+786t-1512=1440$$

Add 1512 to both sides.

$$-70t^{2}+786t=1440+1512$$

Add 1440 and 1512 to get 2952.

$$-70t^{2}+786t=2952$$

Divide both sides by -70.

$$\frac{-70t^{2}+786t}{-70}=\frac{2952}{-70}$$

Dividing by -70 undoes the multiplication by -70.

$$t^{2}+\frac{786}{-70}t=\frac{2952}{-70}$$

Reduce the fraction \frac{786}{-70} to lowest terms by extracting and canceling out 2.

$$t^{2}-\frac{393}{35}t=\frac{2952}{-70}$$

Reduce the fraction \frac{2952}{-70} to lowest terms by extracting and canceling out 2.

$$t^{2}-\frac{393}{35}t=-\frac{1476}{35}$$

Divide -\frac{393}{35}, the coefficient of the x term, by 2 to get -\frac{393}{70}. Then add the square of -\frac{393}{70} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{393}{35}t+\left(-\frac{393}{70}\right)^{2}=-\frac{1476}{35}+\left(-\frac{393}{70}\right)^{2}$$

Square -\frac{393}{70} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{393}{35}t+\frac{154449}{4900}=-\frac{1476}{35}+\frac{154449}{4900}$$

Add -\frac{1476}{35} to \frac{154449}{4900} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{393}{35}t+\frac{154449}{4900}=-\frac{52191}{4900}$$

Factor t^{2}-\frac{393}{35}t+\frac{154449}{4900}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{393}{70}\right)^{2}=-\frac{52191}{4900}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{393}{70}\right)^{2}}=\sqrt{-\frac{52191}{4900}}$$

Simplify.

$$t-\frac{393}{70}=\frac{3\sqrt{5799}i}{70} t-\frac{393}{70}=-\frac{3\sqrt{5799}i}{70}$$

Add \frac{393}{70} to both sides of the equation.

$$t=\frac{393+3\sqrt{5799}i}{70} t=\frac{-3\sqrt{5799}i+393}{70}$$