Solve for t rm=t^2-mt

$r m = t^{2} - m t$

Step 1

Rewrite the equation as $t^{2} - m t = r m$ .

$t^{2} - m t = r m$

Step 2

Subtract $r m$ from both sides of the equation.

$t^{2} - m t - r m = 0$

Step 3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values $a = 1$ , $b = - m$ , and $c = - r m$ into the quadratic formula and solve for $t$ .

$\frac{m \pm \sqrt{{(- m)}^{2} - 4 \cdot (1 \cdot (- r m))}}{2 \cdot 1}$

Step 5

Simplify.

Simplify the numerator.

Apply the product rule to $- m$ .

$t = \frac{m \pm \sqrt{{(- 1)}^{2} m^{2} - 4 \cdot 1 \cdot (- 1 r m)}}{2 \cdot 1}$

Raise $- 1$ to the power of $2$ .

$t = \frac{m \pm \sqrt{1 m^{2} - 4 \cdot 1 \cdot (- 1 r m)}}{2 \cdot 1}$

Multiply $m^{2}$ by $1$ .

$t = \frac{m \pm \sqrt{m^{2} - 4 \cdot 1 \cdot (- 1 r m)}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 1$ .

Multiply $- 4$ by $1$ .

$t = \frac{m \pm \sqrt{m^{2} - 4 \cdot (- 1 r m)}}{2 \cdot 1}$

Multiply $- 4$ by $- 1$ .

$t = \frac{m \pm \sqrt{m^{2} + 4 r m}}{2 \cdot 1}$

Factor $m$ out of $m^{2} + 4 r m$ .

Factor $m$ out of $m^{2}$ .

$t = \frac{m \pm \sqrt{m \cdot m + 4 r m}}{2 \cdot 1}$

Factor $m$ out of $4 r m$ .

$t = \frac{m \pm \sqrt{m \cdot m + m (4 r)}}{2 \cdot 1}$

Factor $m$ out of $m \cdot m + m (4 r)$ .

$t = \frac{m \pm \sqrt{m (m + 4 r)}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$t = \frac{m \pm \sqrt{m (m + 4 r)}}{2}$

Step 6

The final answer is the combination of both solutions.

$t = \frac{m + \sqrt{m (m + 4 r)}}{2}$

$t = \frac{m - \sqrt{m (m + 4 r)}}{2}$