Solve for x U=1/2kx^2

$U = \frac{1}{2} k x^{2}$

Step 1

Rewrite the equation as $\frac{1}{2} \cdot (k x^{2}) = U$ .

$\frac{1}{2} \cdot (k x^{2}) = U$

Step 2

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \cdot (k x^{2})) = 2 U$

Step 3

Simplify the left side.

Simplify $2 (\frac{1}{2} \cdot (k x^{2}))$ .

Multiply $\frac{1}{2} (k x^{2})$ .

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

$2 (\frac{k}{2} x^{2}) = 2 U$

Combine $\frac{k}{2}$ and $x^{2}$ .

$2 \frac{k x^{2}}{2} = 2 U$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 \frac{k x^{2}}{2} = 2 U$

Rewrite the expression.

$k x^{2} = 2 U$

Step 4

Divide each term in $k x^{2} = 2 U$ by $k$ and simplify.

Divide each term in $k x^{2} = 2 U$ by $k$ .

$\frac{k x^{2}}{k} = \frac{2 U}{k}$

Simplify the left side.

Cancel the common factor of $k$ .

Cancel the common factor.

$\frac{k x^{2}}{k} = \frac{2 U}{k}$

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

$x^{2} = \frac{2 U}{k}$

Step 5

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

$x = \pm \sqrt{\frac{2 U}{k}}$

Step 6

Simplify $\pm \sqrt{\frac{2 U}{k}}$ .

Rewrite $\sqrt{\frac{2 U}{k}}$ as $\frac{\sqrt{2 U}}{\sqrt{k}}$ .

$x = \pm \frac{\sqrt{2 U}}{\sqrt{k}}$

Multiply $\frac{\sqrt{2 U}}{\sqrt{k}}$ by $\frac{\sqrt{k}}{\sqrt{k}}$ .

$x = \pm \frac{\sqrt{2 U}}{\sqrt{k}} \cdot \frac{\sqrt{k}}{\sqrt{k}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{2 U}}{\sqrt{k}}$ by $\frac{\sqrt{k}}{\sqrt{k}}$ .

$x = \pm \frac{\sqrt{2 U} \sqrt{k}}{\sqrt{k} \sqrt{k}}$

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

$x = \pm \frac{\sqrt{2 U} \sqrt{k}}{{\sqrt{k}}^{1} \sqrt{k}}$

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

$x = \pm \frac{\sqrt{2 U} \sqrt{k}}{{\sqrt{k}}^{1} {\sqrt{k}}^{1}}$

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

$x = \pm \frac{\sqrt{2 U} \sqrt{k}}{{\sqrt{k}}^{1 + 1}}$

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{2 U} \sqrt{k}}{{\sqrt{k}}^{2}}$

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

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

$x = \pm \frac{\sqrt{2 U} \sqrt{k}}{{(k^{\frac{1}{2}})}^{2}}$

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

$x = \pm \frac{\sqrt{2 U} \sqrt{k}}{k^{\frac{1}{2} \cdot 2}}$

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

$x = \pm \frac{\sqrt{2 U} \sqrt{k}}{k^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \pm \frac{\sqrt{2 U} \sqrt{k}}{k^{\frac{2}{2}}}$

Rewrite the expression.

$x = \pm \frac{\sqrt{2 U} \sqrt{k}}{k^{1}}$

Simplify.

$x = \pm \frac{\sqrt{2 U} \sqrt{k}}{k}$

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{2 U k}}{k}$

Step 7

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.

$x = \frac{\sqrt{2 U k}}{k}$

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

$x = - \frac{\sqrt{2 U k}}{k}$

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

$x = \frac{\sqrt{2 U k}}{k}$

$x = - \frac{\sqrt{2 U k}}{k}$

$x = \frac{\sqrt{2 U k}}{k}$

$x = - \frac{\sqrt{2 U k}}{k}$

Solve for x U=1/2kx^2

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