Solve 2t-3t/t+3-t=t-1-2t/10-(t+3)

$$\frac { 2 t - 3 t } { t + 3 - t } = \frac { t - 1 - 2 t } { 10 - ( t + 3 ) }$$

Solve for t

$$t=1$$

$$t=3$$

Steps Using the Quadratic Formula

Variable t cannot be equal to 7 since division by zero is not defined. Multiply both sides of the equation by 3\left(t-7\right), the least common multiple of t+3-t,10-\left(t+3\right).

$$\left(t-7\right)\left(2t-3t\right)=-3\left(t-1-2t\right)$$

Combine 2t and -3t to get -t.

$$\left(t-7\right)\left(-1\right)t=-3\left(t-1-2t\right)$$

Use the distributive property to multiply t-7 by -1.

$$\left(-t+7\right)t=-3\left(t-1-2t\right)$$

Use the distributive property to multiply -t+7 by t.

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

Combine t and -2t to get -t.

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

Use the distributive property to multiply -3 by -t-1.

$$-t^{2}+7t=3t+3$$

Subtract 3t from both sides.

$$-t^{2}+7t-3t=3$$

Combine 7t and -3t to get 4t.

$$-t^{2}+4t=3$$

Subtract 3 from both sides.

$$-t^{2}+4t-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 4 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-4±\sqrt{4^{2}-4\left(-1\right)\left(-3\right)}}{2\left(-1\right)}$$

Square 4.

$$t=\frac{-4±\sqrt{16-4\left(-1\right)\left(-3\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$t=\frac{-4±\sqrt{16+4\left(-3\right)}}{2\left(-1\right)}$$

Multiply 4 times -3.

$$t=\frac{-4±\sqrt{16-12}}{2\left(-1\right)}$$

Add 16 to -12.

$$t=\frac{-4±\sqrt{4}}{2\left(-1\right)}$$

Take the square root of 4.

$$t=\frac{-4±2}{2\left(-1\right)}$$

Multiply 2 times -1.

$$t=\frac{-4±2}{-2}$$

Now solve the equation t=\frac{-4±2}{-2} when ± is plus. Add -4 to 2.

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

Divide -2 by -2.

$$t=1$$

Now solve the equation t=\frac{-4±2}{-2} when ± is minus. Subtract 2 from -4.

$$t=-\frac{6}{-2}$$

Divide -6 by -2.

$$t=3$$

The equation is now solved.

$$t=1 t=3$$

Steps for Completing the Square

Variable t cannot be equal to 7 since division by zero is not defined. Multiply both sides of the equation by 3\left(t-7\right), the least common multiple of t+3-t,10-\left(t+3\right).

$$\left(t-7\right)\left(2t-3t\right)=-3\left(t-1-2t\right)$$

Combine 2t and -3t to get -t.

$$\left(t-7\right)\left(-1\right)t=-3\left(t-1-2t\right)$$

Use the distributive property to multiply t-7 by -1.

$$\left(-t+7\right)t=-3\left(t-1-2t\right)$$

Use the distributive property to multiply -t+7 by t.

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

Combine t and -2t to get -t.

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

Use the distributive property to multiply -3 by -t-1.

$$-t^{2}+7t=3t+3$$

Subtract 3t from both sides.

$$-t^{2}+7t-3t=3$$

Combine 7t and -3t to get 4t.

$$-t^{2}+4t=3$$

Divide both sides by -1.

$$\frac{-t^{2}+4t}{-1}=\frac{3}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$t^{2}+\frac{4}{-1}t=\frac{3}{-1}$$

Divide 4 by -1.

$$t^{2}-4t=\frac{3}{-1}$$

Divide 3 by -1.

$$t^{2}-4t=-3$$

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

$$t^{2}-4t+\left(-2\right)^{2}=-3+\left(-2\right)^{2}$$

Square -2.

$$t^{2}-4t+4=-3+4$$

Add -3 to 4.

$$t^{2}-4t+4=1$$

Factor t^{2}-4t+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}.

$$\left(t-2\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-2\right)^{2}}=\sqrt{1}$$

Simplify.

$$t-2=1 t-2=-1$$

Add 2 to both sides of the equation.

$$t=3 t=1$$