Solve 1-4t/t^2-3t+2-3(t-1)/t^2-8t+12+7t/t^2-6t=0

$$\frac { 1 - 4 t } { t ^ { 2 } - 3 t + 2 } - \frac { 3 ( t - 1 ) } { t ^ { 2 } - 8 t + 12 } + \frac { 7 t } { t ^ { 2 } - 6 t } = 0$$

Solve for t

$$t=-\frac{1}{2}=-0.5$$

Steps Using Factoring

Variable t cannot be equal to any of the values 0,1,2,6 since division by zero is not defined. Multiply both sides of the equation by t\left(t-6\right)\left(t-2\right)\left(t-1\right), the least common multiple of t^{2}-3t+2,t^{2}-8t+12,t^{2}-6t.

$$t\left(t-6\right)\left(1-4t\right)-\left(t-1\right)t\times 3\left(t-1\right)+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Use the distributive property to multiply t by t-6.

$$\left(t^{2}-6t\right)\left(1-4t\right)-\left(t-1\right)t\times 3\left(t-1\right)+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Use the distributive property to multiply t^{2}-6t by 1-4t and combine like terms.

$$25t^{2}-4t^{3}-6t-\left(t-1\right)t\times 3\left(t-1\right)+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Multiply t-1 and t-1 to get \left(t-1\right)^{2}.

$$25t^{2}-4t^{3}-6t-\left(t-1\right)^{2}t\times 3+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(t-1\right)^{2}.

$$25t^{2}-4t^{3}-6t-\left(t^{2}-2t+1\right)t\times 3+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Use the distributive property to multiply t^{2}-2t+1 by t.

$$25t^{2}-4t^{3}-6t-\left(t^{3}-2t^{2}+t\right)\times 3+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Use the distributive property to multiply t^{3}-2t^{2}+t by 3.

$$25t^{2}-4t^{3}-6t-\left(3t^{3}-6t^{2}+3t\right)+\left(t-1\right)\left(t-2\right)\times 7t=0$$

To find the opposite of 3t^{3}-6t^{2}+3t, find the opposite of each term.

$$25t^{2}-4t^{3}-6t-3t^{3}+6t^{2}-3t+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Combine -4t^{3} and -3t^{3} to get -7t^{3}.

$$25t^{2}-7t^{3}-6t+6t^{2}-3t+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Combine 25t^{2} and 6t^{2} to get 31t^{2}.

$$31t^{2}-7t^{3}-6t-3t+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Combine -6t and -3t to get -9t.

$$31t^{2}-7t^{3}-9t+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Use the distributive property to multiply t-1 by t-2 and combine like terms.

$$31t^{2}-7t^{3}-9t+\left(t^{2}-3t+2\right)\times 7t=0$$

Use the distributive property to multiply t^{2}-3t+2 by 7.

$$31t^{2}-7t^{3}-9t+\left(7t^{2}-21t+14\right)t=0$$

Use the distributive property to multiply 7t^{2}-21t+14 by t.

$$31t^{2}-7t^{3}-9t+7t^{3}-21t^{2}+14t=0$$

Combine -7t^{3} and 7t^{3} to get 0.

$$31t^{2}-9t-21t^{2}+14t=0$$

Combine 31t^{2} and -21t^{2} to get 10t^{2}.

$$10t^{2}-9t+14t=0$$

Combine -9t and 14t to get 5t.

$$10t^{2}+5t=0$$

Factor out t.

$$t\left(10t+5\right)=0$$

To find equation solutions, solve t=0 and 10t+5=0.

$$t=0 t=-\frac{1}{2}$$

Variable t cannot be equal to 0.

$$t=-\frac{1}{2}$$

Steps Using the Quadratic Formula

Variable t cannot be equal to any of the values 0,1,2,6 since division by zero is not defined. Multiply both sides of the equation by t\left(t-6\right)\left(t-2\right)\left(t-1\right), the least common multiple of t^{2}-3t+2,t^{2}-8t+12,t^{2}-6t.

$$t\left(t-6\right)\left(1-4t\right)-\left(t-1\right)t\times 3\left(t-1\right)+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Use the distributive property to multiply t by t-6.

$$\left(t^{2}-6t\right)\left(1-4t\right)-\left(t-1\right)t\times 3\left(t-1\right)+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Use the distributive property to multiply t^{2}-6t by 1-4t and combine like terms.

$$25t^{2}-4t^{3}-6t-\left(t-1\right)t\times 3\left(t-1\right)+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Multiply t-1 and t-1 to get \left(t-1\right)^{2}.

$$25t^{2}-4t^{3}-6t-\left(t-1\right)^{2}t\times 3+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(t-1\right)^{2}.

$$25t^{2}-4t^{3}-6t-\left(t^{2}-2t+1\right)t\times 3+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Use the distributive property to multiply t^{2}-2t+1 by t.

$$25t^{2}-4t^{3}-6t-\left(t^{3}-2t^{2}+t\right)\times 3+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Use the distributive property to multiply t^{3}-2t^{2}+t by 3.

$$25t^{2}-4t^{3}-6t-\left(3t^{3}-6t^{2}+3t\right)+\left(t-1\right)\left(t-2\right)\times 7t=0$$

To find the opposite of 3t^{3}-6t^{2}+3t, find the opposite of each term.

$$25t^{2}-4t^{3}-6t-3t^{3}+6t^{2}-3t+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Combine -4t^{3} and -3t^{3} to get -7t^{3}.

$$25t^{2}-7t^{3}-6t+6t^{2}-3t+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Combine 25t^{2} and 6t^{2} to get 31t^{2}.

$$31t^{2}-7t^{3}-6t-3t+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Combine -6t and -3t to get -9t.

$$31t^{2}-7t^{3}-9t+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Use the distributive property to multiply t-1 by t-2 and combine like terms.

$$31t^{2}-7t^{3}-9t+\left(t^{2}-3t+2\right)\times 7t=0$$

Use the distributive property to multiply t^{2}-3t+2 by 7.

$$31t^{2}-7t^{3}-9t+\left(7t^{2}-21t+14\right)t=0$$

Use the distributive property to multiply 7t^{2}-21t+14 by t.

$$31t^{2}-7t^{3}-9t+7t^{3}-21t^{2}+14t=0$$

Combine -7t^{3} and 7t^{3} to get 0.

$$31t^{2}-9t-21t^{2}+14t=0$$

Combine 31t^{2} and -21t^{2} to get 10t^{2}.

$$10t^{2}-9t+14t=0$$

Combine -9t and 14t to get 5t.

$$10t^{2}+5t=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, 5 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-5±\sqrt{5^{2}}}{2\times 10}$$

Take the square root of 5^{2}.

$$t=\frac{-5±5}{2\times 10}$$

Multiply 2 times 10.

$$t=\frac{-5±5}{20}$$

Now solve the equation t=\frac{-5±5}{20} when ± is plus. Add -5 to 5.

$$t=\frac{0}{20}$$

Divide 0 by 20.

$$t=0$$

Now solve the equation t=\frac{-5±5}{20} when ± is minus. Subtract 5 from -5.

$$t=-\frac{10}{20}$$

Reduce the fraction \frac{-10}{20} to lowest terms by extracting and canceling out 10.

$$t=-\frac{1}{2}$$

The equation is now solved.

$$t=0 t=-\frac{1}{2}$$

Variable t cannot be equal to 0.

$$t=-\frac{1}{2}$$

Steps for Completing the Square

Variable t cannot be equal to any of the values 0,1,2,6 since division by zero is not defined. Multiply both sides of the equation by t\left(t-6\right)\left(t-2\right)\left(t-1\right), the least common multiple of t^{2}-3t+2,t^{2}-8t+12,t^{2}-6t.

$$t\left(t-6\right)\left(1-4t\right)-\left(t-1\right)t\times 3\left(t-1\right)+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Use the distributive property to multiply t by t-6.

$$\left(t^{2}-6t\right)\left(1-4t\right)-\left(t-1\right)t\times 3\left(t-1\right)+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Use the distributive property to multiply t^{2}-6t by 1-4t and combine like terms.

$$25t^{2}-4t^{3}-6t-\left(t-1\right)t\times 3\left(t-1\right)+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Multiply t-1 and t-1 to get \left(t-1\right)^{2}.

$$25t^{2}-4t^{3}-6t-\left(t-1\right)^{2}t\times 3+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(t-1\right)^{2}.

$$25t^{2}-4t^{3}-6t-\left(t^{2}-2t+1\right)t\times 3+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Use the distributive property to multiply t^{2}-2t+1 by t.

$$25t^{2}-4t^{3}-6t-\left(t^{3}-2t^{2}+t\right)\times 3+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Use the distributive property to multiply t^{3}-2t^{2}+t by 3.

$$25t^{2}-4t^{3}-6t-\left(3t^{3}-6t^{2}+3t\right)+\left(t-1\right)\left(t-2\right)\times 7t=0$$

To find the opposite of 3t^{3}-6t^{2}+3t, find the opposite of each term.

$$25t^{2}-4t^{3}-6t-3t^{3}+6t^{2}-3t+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Combine -4t^{3} and -3t^{3} to get -7t^{3}.

$$25t^{2}-7t^{3}-6t+6t^{2}-3t+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Combine 25t^{2} and 6t^{2} to get 31t^{2}.

$$31t^{2}-7t^{3}-6t-3t+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Combine -6t and -3t to get -9t.

$$31t^{2}-7t^{3}-9t+\left(t-1\right)\left(t-2\right)\times 7t=0$$

Use the distributive property to multiply t-1 by t-2 and combine like terms.

$$31t^{2}-7t^{3}-9t+\left(t^{2}-3t+2\right)\times 7t=0$$

Use the distributive property to multiply t^{2}-3t+2 by 7.

$$31t^{2}-7t^{3}-9t+\left(7t^{2}-21t+14\right)t=0$$

Use the distributive property to multiply 7t^{2}-21t+14 by t.

$$31t^{2}-7t^{3}-9t+7t^{3}-21t^{2}+14t=0$$

Combine -7t^{3} and 7t^{3} to get 0.

$$31t^{2}-9t-21t^{2}+14t=0$$

Combine 31t^{2} and -21t^{2} to get 10t^{2}.

$$10t^{2}-9t+14t=0$$

Combine -9t and 14t to get 5t.

$$10t^{2}+5t=0$$

Divide both sides by 10.

$$\frac{10t^{2}+5t}{10}=\frac{0}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$t^{2}+\frac{5}{10}t=\frac{0}{10}$$

Reduce the fraction \frac{5}{10} to lowest terms by extracting and canceling out 5.

$$t^{2}+\frac{1}{2}t=\frac{0}{10}$$

Divide 0 by 10.

$$t^{2}+\frac{1}{2}t=0$$

Divide \frac{1}{2}, the coefficient of the x term, by 2 to get \frac{1}{4}. Then add the square of \frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+\frac{1}{2}t+\left(\frac{1}{4}\right)^{2}=\left(\frac{1}{4}\right)^{2}$$

Square \frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$t^{2}+\frac{1}{2}t+\frac{1}{16}=\frac{1}{16}$$

Factor t^{2}+\frac{1}{2}t+\frac{1}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+\frac{1}{4}\right)^{2}=\frac{1}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+\frac{1}{4}\right)^{2}}=\sqrt{\frac{1}{16}}$$

Simplify.

$$t+\frac{1}{4}=\frac{1}{4} t+\frac{1}{4}=-\frac{1}{4}$$

Subtract \frac{1}{4} from both sides of the equation.

$$t=0 t=-\frac{1}{2}$$

Variable t cannot be equal to 0.

$$t=-\frac{1}{2}$$