Solve 5-1.25t/4-t=4-t/2t-5

$$\frac{ 5-1.25t }{ 4-t } = \frac{ 4-t }{ 2t-5 }$$

Solve for t

$$t = \frac{41}{14} = 2\frac{13}{14} \approx 2.928571429$$

Steps Using the Quadratic Formula

Variable t cannot be equal to any of the values \frac{5}{2},4 since division by zero is not defined. Multiply both sides of the equation by \left(t-4\right)\left(2t-5\right), the least common multiple of 4-t,2t-5.

$$\left(5-2t\right)\left(5-1.25t\right)=\left(t-4\right)\left(4-t\right)$$

Use the distributive property to multiply 5-2t by 5-1.25t and combine like terms.

$$25-16.25t+2.5t^{2}=\left(t-4\right)\left(4-t\right)$$

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

$$25-16.25t+2.5t^{2}=8t-t^{2}-16$$

Subtract 8t from both sides.

$$25-16.25t+2.5t^{2}-8t=-t^{2}-16$$

Combine -16.25t and -8t to get -24.25t.

$$25-24.25t+2.5t^{2}=-t^{2}-16$$

Add t^{2} to both sides.

$$25-24.25t+2.5t^{2}+t^{2}=-16$$

Combine 2.5t^{2} and t^{2} to get 3.5t^{2}.

$$25-24.25t+3.5t^{2}=-16$$

Add 16 to both sides.

$$25-24.25t+3.5t^{2}+16=0$$

Add 25 and 16 to get 41.

$$41-24.25t+3.5t^{2}=0$$

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.

$$3.5t^{2}-24.25t+41=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3.5 for a, -24.25 for b, and 41 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-24.25\right)±\sqrt{\left(-24.25\right)^{2}-4\times 3.5\times 41}}{2\times 3.5}$$

Square -24.25 by squaring both the numerator and the denominator of the fraction.

$$t=\frac{-\left(-24.25\right)±\sqrt{588.0625-4\times 3.5\times 41}}{2\times 3.5}$$

Multiply -4 times 3.5.

$$t=\frac{-\left(-24.25\right)±\sqrt{588.0625-14\times 41}}{2\times 3.5}$$

Multiply -14 times 41.

$$t=\frac{-\left(-24.25\right)±\sqrt{588.0625-574}}{2\times 3.5}$$

Add 588.0625 to -574.

$$t=\frac{-\left(-24.25\right)±\sqrt{14.0625}}{2\times 3.5}$$

Take the square root of 14.0625.

$$t=\frac{-\left(-24.25\right)±\frac{15}{4}}{2\times 3.5}$$

The opposite of -24.25 is 24.25.

$$t=\frac{24.25±\frac{15}{4}}{2\times 3.5}$$

Multiply 2 times 3.5.

$$t=\frac{24.25±\frac{15}{4}}{7}$$

Now solve the equation t=\frac{24.25±\frac{15}{4}}{7} when ± is plus. Add 24.25 to \frac{15}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t=\frac{28}{7}$$

Divide 28 by 7.

$$t=4$$

Now solve the equation t=\frac{24.25±\frac{15}{4}}{7} when ± is minus. Subtract \frac{15}{4} from 24.25 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$t=\frac{\frac{41}{2}}{7}$$

Divide \frac{41}{2} by 7.

$$t=\frac{41}{14}$$

The equation is now solved.

$$t=4 t=\frac{41}{14}$$

Variable t cannot be equal to 4.

$$t=\frac{41}{14}$$

Steps for Completing the Square

Variable t cannot be equal to any of the values \frac{5}{2},4 since division by zero is not defined. Multiply both sides of the equation by \left(t-4\right)\left(2t-5\right), the least common multiple of 4-t,2t-5.

$$\left(5-2t\right)\left(5-1.25t\right)=\left(t-4\right)\left(4-t\right)$$

Use the distributive property to multiply 5-2t by 5-1.25t and combine like terms.

$$25-16.25t+2.5t^{2}=\left(t-4\right)\left(4-t\right)$$

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

$$25-16.25t+2.5t^{2}=8t-t^{2}-16$$

Subtract 8t from both sides.

$$25-16.25t+2.5t^{2}-8t=-t^{2}-16$$

Combine -16.25t and -8t to get -24.25t.

$$25-24.25t+2.5t^{2}=-t^{2}-16$$

Add t^{2} to both sides.

$$25-24.25t+2.5t^{2}+t^{2}=-16$$

Combine 2.5t^{2} and t^{2} to get 3.5t^{2}.

$$25-24.25t+3.5t^{2}=-16$$

Subtract 25 from both sides.

$$-24.25t+3.5t^{2}=-16-25$$

Subtract 25 from -16 to get -41.

$$-24.25t+3.5t^{2}=-41$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$3.5t^{2}-24.25t=-41$$

Divide both sides of the equation by 3.5, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{3.5t^{2}-24.25t}{3.5}=-\frac{41}{3.5}$$

Dividing by 3.5 undoes the multiplication by 3.5.

$$t^{2}+\left(-\frac{24.25}{3.5}\right)t=-\frac{41}{3.5}$$

Divide -24.25 by 3.5 by multiplying -24.25 by the reciprocal of 3.5.

$$t^{2}-\frac{97}{14}t=-\frac{41}{3.5}$$

Divide -41 by 3.5 by multiplying -41 by the reciprocal of 3.5.

$$t^{2}-\frac{97}{14}t=-\frac{82}{7}$$

Divide -\frac{97}{14}, the coefficient of the x term, by 2 to get -\frac{97}{28}. Then add the square of -\frac{97}{28} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{97}{14}t+\left(-\frac{97}{28}\right)^{2}=-\frac{82}{7}+\left(-\frac{97}{28}\right)^{2}$$

Square -\frac{97}{28} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{97}{14}t+\frac{9409}{784}=-\frac{82}{7}+\frac{9409}{784}$$

Add -\frac{82}{7} to \frac{9409}{784} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{97}{14}t+\frac{9409}{784}=\frac{225}{784}$$

Factor t^{2}-\frac{97}{14}t+\frac{9409}{784}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{97}{28}\right)^{2}=\frac{225}{784}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{97}{28}\right)^{2}}=\sqrt{\frac{225}{784}}$$

Simplify.

$$t-\frac{97}{28}=\frac{15}{28} t-\frac{97}{28}=-\frac{15}{28}$$

Add \frac{97}{28} to both sides of the equation.

$$t=4 t=\frac{41}{14}$$

Variable t cannot be equal to 4.

$$t=\frac{41}{14}$$