Solve 10t+10/t=12t+48/t+2

$$\frac { 10 t + 10 } { t } = \frac { 12 t + 48 } { t + 2 }$$

Solve for t

$$t=-10$$

$$t=1$$

Steps Using Factoring By Grouping

Variable t cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by t\left(t+2\right), the least common multiple of t,t+2.

$$\left(t+2\right)\left(10t+10\right)=t\left(12t+48\right)$$

Use the distributive property to multiply t+2 by 10t+10 and combine like terms.

$$10t^{2}+30t+20=t\left(12t+48\right)$$

Use the distributive property to multiply t by 12t+48.

$$10t^{2}+30t+20=12t^{2}+48t$$

Subtract 12t^{2} from both sides.

$$10t^{2}+30t+20-12t^{2}=48t$$

Combine 10t^{2} and -12t^{2} to get -2t^{2}.

$$-2t^{2}+30t+20=48t$$

Subtract 48t from both sides.

$$-2t^{2}+30t+20-48t=0$$

Combine 30t and -48t to get -18t.

$$-2t^{2}-18t+20=0$$

Divide both sides by 2.

$$-t^{2}-9t+10=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -t^{2}+at+bt+10. To find a and b, set up a system to be solved.

$$a+b=-9 ab=-10=-10$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -10.

$$1,-10 2,-5$$

Calculate the sum for each pair.

$$1-10=-9 2-5=-3$$

The solution is the pair that gives sum -9.

$$a=1 b=-10$$

Rewrite -t^{2}-9t+10 as \left(-t^{2}+t\right)+\left(-10t+10\right).

$$\left(-t^{2}+t\right)+\left(-10t+10\right)$$

Factor out t in the first and 10 in the second group.

$$t\left(-t+1\right)+10\left(-t+1\right)$$

Factor out common term -t+1 by using distributive property.

$$\left(-t+1\right)\left(t+10\right)$$

To find equation solutions, solve -t+1=0 and t+10=0.

$$t=1 t=-10$$

Steps Using the Quadratic Formula

Variable t cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by t\left(t+2\right), the least common multiple of t,t+2.

$$\left(t+2\right)\left(10t+10\right)=t\left(12t+48\right)$$

Use the distributive property to multiply t+2 by 10t+10 and combine like terms.

$$10t^{2}+30t+20=t\left(12t+48\right)$$

Use the distributive property to multiply t by 12t+48.

$$10t^{2}+30t+20=12t^{2}+48t$$

Subtract 12t^{2} from both sides.

$$10t^{2}+30t+20-12t^{2}=48t$$

Combine 10t^{2} and -12t^{2} to get -2t^{2}.

$$-2t^{2}+30t+20=48t$$

Subtract 48t from both sides.

$$-2t^{2}+30t+20-48t=0$$

Combine 30t and -48t to get -18t.

$$-2t^{2}-18t+20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -18 for b, and 20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\left(-2\right)\times 20}}{2\left(-2\right)}$$

Square -18.

$$t=\frac{-\left(-18\right)±\sqrt{324-4\left(-2\right)\times 20}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$t=\frac{-\left(-18\right)±\sqrt{324+8\times 20}}{2\left(-2\right)}$$

Multiply 8 times 20.

$$t=\frac{-\left(-18\right)±\sqrt{324+160}}{2\left(-2\right)}$$

Add 324 to 160.

$$t=\frac{-\left(-18\right)±\sqrt{484}}{2\left(-2\right)}$$

Take the square root of 484.

$$t=\frac{-\left(-18\right)±22}{2\left(-2\right)}$$

The opposite of -18 is 18.

$$t=\frac{18±22}{2\left(-2\right)}$$

Multiply 2 times -2.

$$t=\frac{18±22}{-4}$$

Now solve the equation t=\frac{18±22}{-4} when ± is plus. Add 18 to 22.

$$t=\frac{40}{-4}$$

Divide 40 by -4.

$$t=-10$$

Now solve the equation t=\frac{18±22}{-4} when ± is minus. Subtract 22 from 18.

$$t=-\frac{4}{-4}$$

Divide -4 by -4.

$$t=1$$

The equation is now solved.

$$t=-10 t=1$$

Steps for Completing the Square

Variable t cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by t\left(t+2\right), the least common multiple of t,t+2.

$$\left(t+2\right)\left(10t+10\right)=t\left(12t+48\right)$$

Use the distributive property to multiply t+2 by 10t+10 and combine like terms.

$$10t^{2}+30t+20=t\left(12t+48\right)$$

Use the distributive property to multiply t by 12t+48.

$$10t^{2}+30t+20=12t^{2}+48t$$

Subtract 12t^{2} from both sides.

$$10t^{2}+30t+20-12t^{2}=48t$$

Combine 10t^{2} and -12t^{2} to get -2t^{2}.

$$-2t^{2}+30t+20=48t$$

Subtract 48t from both sides.

$$-2t^{2}+30t+20-48t=0$$

Combine 30t and -48t to get -18t.

$$-2t^{2}-18t+20=0$$

Subtract 20 from both sides. Anything subtracted from zero gives its negation.

$$-2t^{2}-18t=-20$$

Divide both sides by -2.

$$\frac{-2t^{2}-18t}{-2}=-\frac{20}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$t^{2}+\left(-\frac{18}{-2}\right)t=-\frac{20}{-2}$$

Divide -18 by -2.

$$t^{2}+9t=-\frac{20}{-2}$$

Divide -20 by -2.

$$t^{2}+9t=10$$

Divide 9, the coefficient of the x term, by 2 to get \frac{9}{2}. Then add the square of \frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+9t+\left(\frac{9}{2}\right)^{2}=10+\left(\frac{9}{2}\right)^{2}$$

Square \frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$t^{2}+9t+\frac{81}{4}=10+\frac{81}{4}$$

Add 10 to \frac{81}{4}.

$$t^{2}+9t+\frac{81}{4}=\frac{121}{4}$$

Factor t^{2}+9t+\frac{81}{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}.

$$\left(t+\frac{9}{2}\right)^{2}=\frac{121}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+\frac{9}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Simplify.

$$t+\frac{9}{2}=\frac{11}{2} t+\frac{9}{2}=-\frac{11}{2}$$

Subtract \frac{9}{2} from both sides of the equation.

$$t=1 t=-10$$