Precalculus Examples

${(x^{2} + y^{2} - 7 y)}^{2} = 49 x^{2} + 49 y^{2}$

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 $x$ .

${(x^{2} + {(0)}^{2} - 7 \cdot 0)}^{2} = 49 x^{2} + 49 {(0)}^{2}$

Step 1.2

Solve the equation.

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Step 1.2.1

Simplify ${(x^{2} + {(0)}^{2} - 7 \cdot 0)}^{2}$ .

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Step 1.2.1.1

Rewrite.

$0 + 0 + {(x^{2} + {(0)}^{2} - 7 \cdot 0)}^{2} = 49 x^{2} + 49 {(0)}^{2}$

Step 1.2.1.2

Simplify by adding zeros.

${(x^{2} + {(0)}^{2} - 7 \cdot 0)}^{2} = 49 x^{2} + 49 {(0)}^{2}$

Step 1.2.1.3

Simplify each term.

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Step 1.2.1.3.1

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

${(x^{2} + 0 - 7 \cdot 0)}^{2} = 49 x^{2} + 49 {(0)}^{2}$

Step 1.2.1.3.2

Multiply $- 7$ by $0$ .

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

Step 1.2.1.4

Simplify by adding terms.

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Step 1.2.1.4.1

Combine the opposite terms in $x^{2} + 0 + 0$ .

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Step 1.2.1.4.1.1

Add $x^{2}$ and $0$ .

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

Step 1.2.1.4.1.2

Add $x^{2}$ and $0$ .

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

Step 1.2.1.4.2

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 1.2.1.4.2.1

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

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

Step 1.2.1.4.2.2

Multiply $2$ by $2$ .

$x^{4} = 49 x^{2} + 49 {(0)}^{2}$

Step 1.2.2

Simplify $49 x^{2} + 49 {(0)}^{2}$ .

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Step 1.2.2.1

Simplify each term.

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Step 1.2.2.1.1

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

$x^{4} = 49 x^{2} + 49 \cdot 0$

Step 1.2.2.1.2

Multiply $49$ by $0$ .

$x^{4} = 49 x^{2} + 0$

Step 1.2.2.2

Add $49 x^{2}$ and $0$ .

$x^{4} = 49 x^{2}$

Step 1.2.3

Subtract $49 x^{2}$ from both sides of the equation.

$x^{4} - 49 x^{2} = 0$

Step 1.2.4

Factor the left side of the equation.

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Step 1.2.4.1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

${(x^{2})}^{2} - 49 x^{2} = 0$

Step 1.2.4.2

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$u^{2} - 49 u = 0$

Step 1.2.4.3

Factor $u$ out of $u^{2} - 49 u$ .

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Step 1.2.4.3.1

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

$u \cdot u - 49 u = 0$

Step 1.2.4.3.2

Factor $u$ out of $- 49 u$ .

$u \cdot u + u \cdot - 49 = 0$

Step 1.2.4.3.3

Factor $u$ out of $u \cdot u + u \cdot - 49$ .

$u (u - 49) = 0$

Step 1.2.4.4

Replace all occurrences of $u$ with $x^{2}$ .

$x^{2} (x^{2} - 49) = 0$

Step 1.2.5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{2} = 0$

$x^{2} - 49 = 0$

Step 1.2.6

Set $x^{2}$ equal to $0$ and solve for $x$ .

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Step 1.2.6.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 1.2.6.2

Solve $x^{2} = 0$ for $x$ .

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Step 1.2.6.2.1

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

$x = \pm \sqrt{0}$

Step 1.2.6.2.2

Simplify $\pm \sqrt{0}$ .

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Step 1.2.6.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 1.2.6.2.2.2

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

$x = \pm 0$

Step 1.2.6.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.2.7

Set $x^{2} - 49$ equal to $0$ and solve for $x$ .

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Step 1.2.7.1

Set $x^{2} - 49$ equal to $0$ .

$x^{2} - 49 = 0$

Step 1.2.7.2

Solve $x^{2} - 49 = 0$ for $x$ .

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Step 1.2.7.2.1

Add $49$ to both sides of the equation.

$x^{2} = 49$

Step 1.2.7.2.2

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

$x = \pm \sqrt{49}$

Step 1.2.7.2.3

Simplify $\pm \sqrt{49}$ .

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Step 1.2.7.2.3.1

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

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

Step 1.2.7.2.3.2

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

$x = \pm 7$

Step 1.2.7.2.4

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

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Step 1.2.7.2.4.1

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

$x = 7$

Step 1.2.7.2.4.2

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

$x = - 7$

Step 1.2.7.2.4.3

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

$x = 7, - 7$

Step 1.2.8

The final solution is all the values that make $x^{2} (x^{2} - 49) = 0$ true.

$x = 0, 7, - 7$

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

${({(0)}^{2} + y^{2} - 7 y)}^{2} = 49 {(0)}^{2} + 49 y^{2}$

Step 2.2

Solve the equation.

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Step 2.2.1

Simplify ${({(0)}^{2} + y^{2} - 7 y)}^{2}$ .

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Step 2.2.1.1

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

${(0 + y^{2} - 7 y)}^{2} = 49 {(0)}^{2} + 49 y^{2}$

Step 2.2.1.2

Add $0$ and $y^{2}$ .

${(y^{2} - 7 y)}^{2} = 49 {(0)}^{2} + 49 y^{2}$

Step 2.2.2

Simplify $49 {(0)}^{2} + 49 y^{2}$ .

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

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

${(y^{2} - 7 y)}^{2} = 49 \cdot 0 + 49 y^{2}$

Step 2.2.2.1.2

Multiply $49$ by $0$ .

${(y^{2} - 7 y)}^{2} = 0 + 49 y^{2}$

Step 2.2.2.2

Add $0$ and $49 y^{2}$ .

${(y^{2} - 7 y)}^{2} = 49 y^{2}$

Step 2.2.3

Move all terms containing $y$ to the left side of the equation.

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Step 2.2.3.1

Subtract $49 y^{2}$ from both sides of the equation.

${(y^{2} - 7 y)}^{2} - 49 y^{2} = 0$

Step 2.2.3.2

Simplify each term.

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Step 2.2.3.2.1

Rewrite ${(y^{2} - 7 y)}^{2}$ as $(y^{2} - 7 y) (y^{2} - 7 y)$ .

$(y^{2} - 7 y) (y^{2} - 7 y) - 49 y^{2} = 0$

Step 2.2.3.2.2

Expand $(y^{2} - 7 y) (y^{2} - 7 y)$ using the FOIL Method.

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Step 2.2.3.2.2.1

Apply the distributive property.

$y^{2} (y^{2} - 7 y) - 7 y (y^{2} - 7 y) - 49 y^{2} = 0$

Step 2.2.3.2.2.2

Apply the distributive property.

$y^{2} y^{2} + y^{2} (- 7 y) - 7 y (y^{2} - 7 y) - 49 y^{2} = 0$

Step 2.2.3.2.2.3

Apply the distributive property.

$y^{2} y^{2} + y^{2} (- 7 y) - 7 y \cdot y^{2} - 7 y (- 7 y) - 49 y^{2} = 0$

Step 2.2.3.2.3

Simplify and combine like terms.

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Step 2.2.3.2.3.1

Simplify each term.

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Step 2.2.3.2.3.1.1

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

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Step 2.2.3.2.3.1.1.1

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

$y^{2 + 2} + y^{2} (- 7 y) - 7 y \cdot y^{2} - 7 y (- 7 y) - 49 y^{2} = 0$

Step 2.2.3.2.3.1.1.2

Add $2$ and $2$ .

$y^{4} + y^{2} (- 7 y) - 7 y \cdot y^{2} - 7 y (- 7 y) - 49 y^{2} = 0$

Step 2.2.3.2.3.1.2

Rewrite using the commutative property of multiplication.

$y^{4} - 7 y^{2} y - 7 y \cdot y^{2} - 7 y (- 7 y) - 49 y^{2} = 0$

Step 2.2.3.2.3.1.3

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 2.2.3.2.3.1.3.1

Move $y$ .

$y^{4} - 7 (y \cdot y^{2}) - 7 y \cdot y^{2} - 7 y (- 7 y) - 49 y^{2} = 0$

Step 2.2.3.2.3.1.3.2

Multiply $y$ by $y^{2}$ .

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Step 2.2.3.2.3.1.3.2.1

Raise $y$ to the power of $1$ .

$y^{4} - 7 (y^{1} y^{2}) - 7 y \cdot y^{2} - 7 y (- 7 y) - 49 y^{2} = 0$

Step 2.2.3.2.3.1.3.2.2

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

$y^{4} - 7 y^{1 + 2} - 7 y \cdot y^{2} - 7 y (- 7 y) - 49 y^{2} = 0$

Step 2.2.3.2.3.1.3.3

Add $1$ and $2$ .

$y^{4} - 7 y^{3} - 7 y \cdot y^{2} - 7 y (- 7 y) - 49 y^{2} = 0$

Step 2.2.3.2.3.1.4

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 2.2.3.2.3.1.4.1

Move $y^{2}$ .

$y^{4} - 7 y^{3} - 7 (y^{2} y) - 7 y (- 7 y) - 49 y^{2} = 0$

Step 2.2.3.2.3.1.4.2

Multiply $y^{2}$ by $y$ .

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Step 2.2.3.2.3.1.4.2.1

Raise $y$ to the power of $1$ .

$y^{4} - 7 y^{3} - 7 (y^{2} y^{1}) - 7 y (- 7 y) - 49 y^{2} = 0$

Step 2.2.3.2.3.1.4.2.2

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

$y^{4} - 7 y^{3} - 7 y^{2 + 1} - 7 y (- 7 y) - 49 y^{2} = 0$

Step 2.2.3.2.3.1.4.3

Add $2$ and $1$ .

$y^{4} - 7 y^{3} - 7 y^{3} - 7 y (- 7 y) - 49 y^{2} = 0$

Step 2.2.3.2.3.1.5

Rewrite using the commutative property of multiplication.

$y^{4} - 7 y^{3} - 7 y^{3} - 7 \cdot - 7 y \cdot y - 49 y^{2} = 0$

Step 2.2.3.2.3.1.6

Multiply $y$ by $y$ by adding the exponents.

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Step 2.2.3.2.3.1.6.1

Move $y$ .

$y^{4} - 7 y^{3} - 7 y^{3} - 7 \cdot - 7 (y \cdot y) - 49 y^{2} = 0$

Step 2.2.3.2.3.1.6.2

Multiply $y$ by $y$ .

$y^{4} - 7 y^{3} - 7 y^{3} - 7 \cdot - 7 y^{2} - 49 y^{2} = 0$

Step 2.2.3.2.3.1.7

Multiply $- 7$ by $- 7$ .

$y^{4} - 7 y^{3} - 7 y^{3} + 49 y^{2} - 49 y^{2} = 0$

Step 2.2.3.2.3.2

Subtract $7 y^{3}$ from $- 7 y^{3}$ .

$y^{4} - 14 y^{3} + 49 y^{2} - 49 y^{2} = 0$

Step 2.2.3.3

Combine the opposite terms in $y^{4} - 14 y^{3} + 49 y^{2} - 49 y^{2}$ .

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Step 2.2.3.3.1

Subtract $49 y^{2}$ from $49 y^{2}$ .

$y^{4} - 14 y^{3} + 0 = 0$

Step 2.2.3.3.2

Add $y^{4} - 14 y^{3}$ and $0$ .

$y^{4} - 14 y^{3} = 0$

Step 2.2.4

Factor $y^{3}$ out of $y^{4} - 14 y^{3}$ .

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Step 2.2.4.1

Factor $y^{3}$ out of $y^{4}$ .

$y^{3} y - 14 y^{3} = 0$

Step 2.2.4.2

Factor $y^{3}$ out of $- 14 y^{3}$ .

$y^{3} y + y^{3} \cdot - 14 = 0$

Step 2.2.4.3

Factor $y^{3}$ out of $y^{3} y + y^{3} \cdot - 14$ .

$y^{3} (y - 14) = 0$

Step 2.2.5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y^{3} = 0$

$y - 14 = 0$

Step 2.2.6

Set $y^{3}$ equal to $0$ and solve for $y$ .

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Step 2.2.6.1

Set $y^{3}$ equal to $0$ .

$y^{3} = 0$

Step 2.2.6.2

Solve $y^{3} = 0$ for $y$ .

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Step 2.2.6.2.1

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

$y = \sqrt[3]{0}$

Step 2.2.6.2.2

Simplify $\sqrt[3]{0}$ .

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Step 2.2.6.2.2.1

Rewrite $0$ as $0^{3}$ .

$y = \sqrt[3]{0^{3}}$

Step 2.2.6.2.2.2

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

$y = 0$

Step 2.2.7

Set $y - 14$ equal to $0$ and solve for $y$ .

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Step 2.2.7.1

Set $y - 14$ equal to $0$ .

$y - 14 = 0$

Step 2.2.7.2

Add $14$ to both sides of the equation.

$y = 14$

Step 2.2.8

The final solution is all the values that make $y^{3} (y - 14) = 0$ true.

$y = 0, 14$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 0), (0, 14)$

Step 3

List the intersections.

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

y-intercept(s): $(0, 0), (0, 14)$

Step 4