Calculus Examples

$4 x^{2} - y^{2} = 7$

Step 1

Add $y^{2}$ to both sides of the equation.

$4 x^{2} = 7 + y^{2}$

Step 2

Divide each term in $4 x^{2} = 7 + y^{2}$ by $4$ and simplify.

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Step 2.1

Divide each term in $4 x^{2} = 7 + y^{2}$ by $4$ .

$\frac{4 x^{2}}{4} = \frac{7}{4} + \frac{y^{2}}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{4 x^{2}}{4} = \frac{7}{4} + \frac{y^{2}}{4}$

Step 2.2.1.2

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

$x^{2} = \frac{7}{4} + \frac{y^{2}}{4}$

Step 3

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

$x = \pm \sqrt{\frac{7}{4} + \frac{y^{2}}{4}}$

Step 4

Simplify $\pm \sqrt{\frac{7}{4} + \frac{y^{2}}{4}}$ .

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Step 4.1

Combine the numerators over the common denominator.

$x = \pm \sqrt{\frac{7 + y^{2}}{4}}$

Step 4.2

Rewrite $\frac{7 + y^{2}}{4}$ as ${(\frac{1}{2})}^{2} (7 + y^{2})$ .

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Step 4.2.1

Factor the perfect power $1^{2}$ out of $7 + y^{2}$ .

$x = \pm \sqrt{\frac{1^{2} (7 + y^{2})}{4}}$

Step 4.2.2

Factor the perfect power $2^{2}$ out of $4$ .

$x = \pm \sqrt{\frac{1^{2} (7 + y^{2})}{2^{2} \cdot 1}}$

Step 4.2.3

Rearrange the fraction $\frac{1^{2} (7 + y^{2})}{2^{2} \cdot 1}$ .

$x = \pm \sqrt{{(\frac{1}{2})}^{2} (7 + y^{2})}$

Step 4.3

Pull terms out from under the radical.

$x = \pm \frac{1}{2} \sqrt{7 + y^{2}}$

Step 4.4

Combine $\frac{1}{2}$ and $\sqrt{7 + y^{2}}$ .

$x = \pm \frac{\sqrt{7 + y^{2}}}{2}$

Step 5

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

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Step 5.1

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

$x = \frac{\sqrt{7 + y^{2}}}{2}$

Step 5.2

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

$x = - \frac{\sqrt{7 + y^{2}}}{2}$

Step 5.3

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

$x = \frac{\sqrt{7 + y^{2}}}{2}$

$x = - \frac{\sqrt{7 + y^{2}}}{2}$

$x = \frac{\sqrt{7 + y^{2}}}{2}$

$x = - \frac{\sqrt{7 + y^{2}}}{2}$