Solve t+5/t=-6

$$t + \frac { 5 } { t } = - 6$$

Solve for t

$$t=-5$$

$$t=-1$$

Steps Using Factoring

Variable t cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by t.

$$tt+5=-6t$$

Multiply t and t to get t^{2}.

$$t^{2}+5=-6t$$

Add 6t to both sides.

$$t^{2}+5+6t=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$t^{2}+6t+5=0$$

To solve the equation, factor t^{2}+6t+5 using formula t^{2}+\left(a+b\right)t+ab=\left(t+a\right)\left(t+b\right). To find a and b, set up a system to be solved.

$$a+b=6 ab=5$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=1 b=5$$

Rewrite factored expression \left(t+a\right)\left(t+b\right) using the obtained values.

$$\left(t+1\right)\left(t+5\right)$$

To find equation solutions, solve t+1=0 and t+5=0.

$$t=-1 t=-5$$

Steps Using Factoring By Grouping

Variable t cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by t.

$$tt+5=-6t$$

Multiply t and t to get t^{2}.

$$t^{2}+5=-6t$$

Add 6t to both sides.

$$t^{2}+5+6t=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$t^{2}+6t+5=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as t^{2}+at+bt+5. To find a and b, set up a system to be solved.

$$a+b=6 ab=1\times 5=5$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=1 b=5$$

Rewrite t^{2}+6t+5 as \left(t^{2}+t\right)+\left(5t+5\right).

$$\left(t^{2}+t\right)+\left(5t+5\right)$$

Factor out t in the first and 5 in the second group.

$$t\left(t+1\right)+5\left(t+1\right)$$

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

$$\left(t+1\right)\left(t+5\right)$$

To find equation solutions, solve t+1=0 and t+5=0.

$$t=-1 t=-5$$

Steps Using the Quadratic Formula

Variable t cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by t.

$$tt+5=-6t$$

Multiply t and t to get t^{2}.

$$t^{2}+5=-6t$$

Add 6t to both sides.

$$t^{2}+5+6t=0$$

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.

$$t^{2}+6t+5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 6 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-6±\sqrt{6^{2}-4\times 5}}{2}$$

Square 6.

$$t=\frac{-6±\sqrt{36-4\times 5}}{2}$$

Multiply -4 times 5.

$$t=\frac{-6±\sqrt{36-20}}{2}$$

Add 36 to -20.

$$t=\frac{-6±\sqrt{16}}{2}$$

Take the square root of 16.

$$t=\frac{-6±4}{2}$$

Now solve the equation t=\frac{-6±4}{2} when ± is plus. Add -6 to 4.

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

Divide -2 by 2.

$$t=-1$$

Now solve the equation t=\frac{-6±4}{2} when ± is minus. Subtract 4 from -6.

$$t=-\frac{10}{2}$$

Divide -10 by 2.

$$t=-5$$

The equation is now solved.

$$t=-1 t=-5$$

Steps for Completing the Square

Variable t cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by t.

$$tt+5=-6t$$

Multiply t and t to get t^{2}.

$$t^{2}+5=-6t$$

Add 6t to both sides.

$$t^{2}+5+6t=0$$

Subtract 5 from both sides. Anything subtracted from zero gives its negation.

$$t^{2}+6t=-5$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+6t+3^{2}=-5+3^{2}$$

Square 3.

$$t^{2}+6t+9=-5+9$$

Add -5 to 9.

$$t^{2}+6t+9=4$$

Factor t^{2}+6t+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$t+3=2 t+3=-2$$

Subtract 3 from both sides of the equation.

$$t=-1 t=-5$$