Solve for t at^2+uy=h

$a t^{2} + u y = h$

Step 1

Subtract $u y$ from both sides of the equation.

$a t^{2} = h - u y$

Step 2

Divide each term in $a t^{2} = h - u y$ by $a$ and simplify.

Divide each term in $a t^{2} = h - u y$ by $a$ .

$\frac{a t^{2}}{a} = \frac{h}{a} + \frac{- u y}{a}$

Simplify the left side.

Cancel the common factor of $a$ .

Cancel the common factor.

$\frac{a t^{2}}{a} = \frac{h}{a} + \frac{- u y}{a}$

Divide $t^{2}$ by $1$ .

$t^{2} = \frac{h}{a} + \frac{- u y}{a}$

Simplify the right side.

Move the negative in front of the fraction.

$t^{2} = \frac{h}{a} - \frac{u y}{a}$

Step 3

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$t = \pm \sqrt{\frac{h}{a} - \frac{u y}{a}}$

Step 4

Simplify $\pm \sqrt{\frac{h}{a} - \frac{u y}{a}}$ .

Combine the numerators over the common denominator.

$t = \pm \sqrt{\frac{h - u y}{a}}$

Rewrite $\sqrt{\frac{h - u y}{a}}$ as $\frac{\sqrt{h - u y}}{\sqrt{a}}$ .

$t = \pm \frac{\sqrt{h - u y}}{\sqrt{a}}$

Multiply $\frac{\sqrt{h - u y}}{\sqrt{a}}$ by $\frac{\sqrt{a}}{\sqrt{a}}$ .

$t = \pm \frac{\sqrt{h - u y}}{\sqrt{a}} \cdot \frac{\sqrt{a}}{\sqrt{a}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{h - u y}}{\sqrt{a}}$ by $\frac{\sqrt{a}}{\sqrt{a}}$ .

$t = \pm \frac{\sqrt{h - u y} \sqrt{a}}{\sqrt{a} \sqrt{a}}$

Raise $\sqrt{a}$ to the power of $1$ .

$t = \pm \frac{\sqrt{h - u y} \sqrt{a}}{{\sqrt{a}}^{1} \sqrt{a}}$

Raise $\sqrt{a}$ to the power of $1$ .

$t = \pm \frac{\sqrt{h - u y} \sqrt{a}}{{\sqrt{a}}^{1} {\sqrt{a}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$t = \pm \frac{\sqrt{h - u y} \sqrt{a}}{{\sqrt{a}}^{1 + 1}}$

Add $1$ and $1$ .

$t = \pm \frac{\sqrt{h - u y} \sqrt{a}}{{\sqrt{a}}^{2}}$

Rewrite ${\sqrt{a}}^{2}$ as $a$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a}$ as $a^{\frac{1}{2}}$ .

$t = \pm \frac{\sqrt{h - u y} \sqrt{a}}{{(a^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$t = \pm \frac{\sqrt{h - u y} \sqrt{a}}{a^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$t = \pm \frac{\sqrt{h - u y} \sqrt{a}}{a^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$t = \pm \frac{\sqrt{h - u y} \sqrt{a}}{a^{\frac{2}{2}}}$

Rewrite the expression.

$t = \pm \frac{\sqrt{h - u y} \sqrt{a}}{a^{1}}$

Simplify.

$t = \pm \frac{\sqrt{h - u y} \sqrt{a}}{a}$

Combine using the product rule for radicals.

$t = \pm \frac{\sqrt{(h - u y) a}}{a}$

Reorder factors in $\pm \frac{\sqrt{(h - u y) a}}{a}$ .

$t = \pm \frac{\sqrt{a (h - u y)}}{a}$

Step 5

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$t = \frac{\sqrt{a (h - u y)}}{a}$

Next, use the negative value of the $\pm$ to find the second solution.

$t = - \frac{\sqrt{a (h - u y)}}{a}$

The complete solution is the result of both the positive and negative portions of the solution.

$t = \frac{\sqrt{a (h - u y)}}{a}$

$t = - \frac{\sqrt{a (h - u y)}}{a}$

$t = \frac{\sqrt{a (h - u y)}}{a}$

$t = - \frac{\sqrt{a (h - u y)}}{a}$

Solve for t at^2+uy=h

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