Precalculus Examples

$y^{2} = 49 + 25 x^{2}$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

${(0)}^{2} = 49 + 25 x^{2}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $49 + 25 x^{2} = {(0)}^{2}$ .

$49 + 25 x^{2} = {(0)}^{2}$

Step 1.2.2

Raising $0$ to any positive power yields $0$ .

$49 + 25 x^{2} = 0$

Step 1.2.3

Subtract $49$ from both sides of the equation.

$25 x^{2} = - 49$

Step 1.2.4

Divide each term in $25 x^{2} = - 49$ by $25$ and simplify.

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Step 1.2.4.1

Divide each term in $25 x^{2} = - 49$ by $25$ .

$\frac{25 x^{2}}{25} = \frac{- 49}{25}$

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

Cancel the common factor of $25$ .

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Step 1.2.4.2.1.1

Cancel the common factor.

$\frac{25 x^{2}}{25} = \frac{- 49}{25}$

Step 1.2.4.2.1.2

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

$x^{2} = \frac{- 49}{25}$

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{49}{25}$

Step 1.2.5

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

$x = \pm \sqrt{- \frac{49}{25}}$

Step 1.2.6

Simplify $\pm \sqrt{- \frac{49}{25}}$ .

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Step 1.2.6.1

Rewrite $- \frac{49}{25}$ as ${(\frac{7 i}{5})}^{2}$ .

$x = \pm \sqrt{{(\frac{7 i}{5})}^{2}}$

Step 1.2.6.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm \frac{7 i}{5}$

Step 1.2.7

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

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Step 1.2.7.1

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

$x = \frac{7 i}{5}$

Step 1.2.7.2

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

$x = - \frac{7 i}{5}$

Step 1.2.7.3

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

$x = \frac{7 i}{5}, - \frac{7 i}{5}$

Step 1.3

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

x-intercept(s): $None$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y^{2} = 49 + 25 {(0)}^{2}$

Step 2.2

Solve the equation.

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Step 2.2.1

Raising $0$ to any positive power yields $0$ .

$y^{2} = 49 + 25 \cdot 0$

Step 2.2.2

Multiply $25$ by $0$ .

$y^{2} = 49 + 0$

Step 2.2.3

Add $49$ and $0$ .

$y^{2} = 49$

Step 2.2.4

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

$y = \pm \sqrt{49}$

Step 2.2.5

Simplify $\pm \sqrt{49}$ .

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Step 2.2.5.1

Rewrite $49$ as $7^{2}$ .

$y = \pm \sqrt{7^{2}}$

Step 2.2.5.2

Pull terms out from under the radical, assuming positive real numbers.

$y = \pm 7$

Step 2.2.6

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

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Step 2.2.6.1

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

$y = 7$

Step 2.2.6.2

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

$y = - 7$

Step 2.2.6.3

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

$y = 7, - 7$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 7), (0, - 7)$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, 7), (0, - 7)$

Step 4