Solve 7/t+1/t+3=1/4

$$\frac { 7 } { t } + \frac { 1 } { t + 3 } = \frac { 1 } { 4 }$$

Solve for t

$$t = \frac{\sqrt{1177} + 29}{2} \approx 31.653716798$$

$$t=\frac{29-\sqrt{1177}}{2}\approx -2.653716798$$

Steps Using the Quadratic Formula

Variable t cannot be equal to any of the values -3,0 since division by zero is not defined. Multiply both sides of the equation by 4t\left(t+3\right), the least common multiple of t,t+3,4.

$$\left(4t+12\right)\times 7+4t=t\left(t+3\right)$$

Use the distributive property to multiply 4t+12 by 7.

$$28t+84+4t=t\left(t+3\right)$$

Combine 28t and 4t to get 32t.

$$32t+84=t\left(t+3\right)$$

Use the distributive property to multiply t by t+3.

$$32t+84=t^{2}+3t$$

Subtract t^{2} from both sides.

$$32t+84-t^{2}=3t$$

Subtract 3t from both sides.

$$32t+84-t^{2}-3t=0$$

Combine 32t and -3t to get 29t.

$$29t+84-t^{2}=0$$

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.

$$-t^{2}+29t+84=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 29 for b, and 84 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-29±\sqrt{29^{2}-4\left(-1\right)\times 84}}{2\left(-1\right)}$$

Square 29.

$$t=\frac{-29±\sqrt{841-4\left(-1\right)\times 84}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$t=\frac{-29±\sqrt{841+4\times 84}}{2\left(-1\right)}$$

Multiply 4 times 84.

$$t=\frac{-29±\sqrt{841+336}}{2\left(-1\right)}$$

Add 841 to 336.

$$t=\frac{-29±\sqrt{1177}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$t=\frac{-29±\sqrt{1177}}{-2}$$

Now solve the equation t=\frac{-29±\sqrt{1177}}{-2} when ± is plus. Add -29 to \sqrt{1177}.

$$t=\frac{\sqrt{1177}-29}{-2}$$

Divide -29+\sqrt{1177} by -2.

$$t=\frac{29-\sqrt{1177}}{2}$$

Now solve the equation t=\frac{-29±\sqrt{1177}}{-2} when ± is minus. Subtract \sqrt{1177} from -29.

$$t=\frac{-\sqrt{1177}-29}{-2}$$

Divide -29-\sqrt{1177} by -2.

$$t=\frac{\sqrt{1177}+29}{2}$$

The equation is now solved.

$$t=\frac{29-\sqrt{1177}}{2} t=\frac{\sqrt{1177}+29}{2}$$

Steps for Completing the Square

Variable t cannot be equal to any of the values -3,0 since division by zero is not defined. Multiply both sides of the equation by 4t\left(t+3\right), the least common multiple of t,t+3,4.

$$\left(4t+12\right)\times 7+4t=t\left(t+3\right)$$

Use the distributive property to multiply 4t+12 by 7.

$$28t+84+4t=t\left(t+3\right)$$

Combine 28t and 4t to get 32t.

$$32t+84=t\left(t+3\right)$$

Use the distributive property to multiply t by t+3.

$$32t+84=t^{2}+3t$$

Subtract t^{2} from both sides.

$$32t+84-t^{2}=3t$$

Subtract 3t from both sides.

$$32t+84-t^{2}-3t=0$$

Combine 32t and -3t to get 29t.

$$29t+84-t^{2}=0$$

Subtract 84 from both sides. Anything subtracted from zero gives its negation.

$$29t-t^{2}=-84$$

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.

$$-t^{2}+29t=-84$$

Divide both sides by -1.

$$\frac{-t^{2}+29t}{-1}=-\frac{84}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$t^{2}+\frac{29}{-1}t=-\frac{84}{-1}$$

Divide 29 by -1.

$$t^{2}-29t=-\frac{84}{-1}$$

Divide -84 by -1.

$$t^{2}-29t=84$$

Divide -29, the coefficient of the x term, by 2 to get -\frac{29}{2}. Then add the square of -\frac{29}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-29t+\left(-\frac{29}{2}\right)^{2}=84+\left(-\frac{29}{2}\right)^{2}$$

Square -\frac{29}{2} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-29t+\frac{841}{4}=84+\frac{841}{4}$$

Add 84 to \frac{841}{4}.

$$t^{2}-29t+\frac{841}{4}=\frac{1177}{4}$$

Factor t^{2}-29t+\frac{841}{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}.

$$\left(t-\frac{29}{2}\right)^{2}=\frac{1177}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{29}{2}\right)^{2}}=\sqrt{\frac{1177}{4}}$$

Simplify.

$$t-\frac{29}{2}=\frac{\sqrt{1177}}{2} t-\frac{29}{2}=-\frac{\sqrt{1177}}{2}$$

Add \frac{29}{2} to both sides of the equation.

$$t=\frac{\sqrt{1177}+29}{2} t=\frac{29-\sqrt{1177}}{2}$$