Algebra Examples

${(x^{2} + y^{2} - 8 x)}^{2} = 64 x^{2} + 64 y^{2}$

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 $x$ .

${(x^{2} + {(0)}^{2} - 8 x)}^{2} = 64 x^{2} + 64 {(0)}^{2}$

Step 1.2

Solve the equation.

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Step 1.2.1

Simplify ${(x^{2} + {(0)}^{2} - 8 x)}^{2}$ .

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Step 1.2.1.1

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

${(x^{2} + 0 - 8 x)}^{2} = 64 x^{2} + 64 {(0)}^{2}$

Step 1.2.1.2

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

${(x^{2} - 8 x)}^{2} = 64 x^{2} + 64 {(0)}^{2}$

Step 1.2.2

Simplify $64 x^{2} + 64 {(0)}^{2}$ .

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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 $0$ .

${(x^{2} - 8 x)}^{2} = 64 x^{2} + 64 \cdot 0$

Step 1.2.2.1.2

Multiply $64$ by $0$ .

${(x^{2} - 8 x)}^{2} = 64 x^{2} + 0$

Step 1.2.2.2

Add $64 x^{2}$ and $0$ .

${(x^{2} - 8 x)}^{2} = 64 x^{2}$

Step 1.2.3

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

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Step 1.2.3.1

Subtract $64 x^{2}$ from both sides of the equation.

${(x^{2} - 8 x)}^{2} - 64 x^{2} = 0$

Step 1.2.3.2

Simplify each term.

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Step 1.2.3.2.1

Rewrite ${(x^{2} - 8 x)}^{2}$ as $(x^{2} - 8 x) (x^{2} - 8 x)$ .

$(x^{2} - 8 x) (x^{2} - 8 x) - 64 x^{2} = 0$

Step 1.2.3.2.2

Expand $(x^{2} - 8 x) (x^{2} - 8 x)$ using the FOIL Method.

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Step 1.2.3.2.2.1

Apply the distributive property.

$x^{2} (x^{2} - 8 x) - 8 x (x^{2} - 8 x) - 64 x^{2} = 0$

Step 1.2.3.2.2.2

Apply the distributive property.

$x^{2} x^{2} + x^{2} (- 8 x) - 8 x (x^{2} - 8 x) - 64 x^{2} = 0$

Step 1.2.3.2.2.3

Apply the distributive property.

$x^{2} x^{2} + x^{2} (- 8 x) - 8 x \cdot x^{2} - 8 x (- 8 x) - 64 x^{2} = 0$

Step 1.2.3.2.3

Simplify and combine like terms.

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Step 1.2.3.2.3.1

Simplify each term.

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Step 1.2.3.2.3.1.1

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

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Step 1.2.3.2.3.1.1.1

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

$x^{2 + 2} + x^{2} (- 8 x) - 8 x \cdot x^{2} - 8 x (- 8 x) - 64 x^{2} = 0$

Step 1.2.3.2.3.1.1.2

Add $2$ and $2$ .

$x^{4} + x^{2} (- 8 x) - 8 x \cdot x^{2} - 8 x (- 8 x) - 64 x^{2} = 0$

Step 1.2.3.2.3.1.2

Rewrite using the commutative property of multiplication.

$x^{4} - 8 x^{2} x - 8 x \cdot x^{2} - 8 x (- 8 x) - 64 x^{2} = 0$

Step 1.2.3.2.3.1.3

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

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Step 1.2.3.2.3.1.3.1

Move $x$ .

$x^{4} - 8 (x \cdot x^{2}) - 8 x \cdot x^{2} - 8 x (- 8 x) - 64 x^{2} = 0$

Step 1.2.3.2.3.1.3.2

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

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Step 1.2.3.2.3.1.3.2.1

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

$x^{4} - 8 (x^{1} x^{2}) - 8 x \cdot x^{2} - 8 x (- 8 x) - 64 x^{2} = 0$

Step 1.2.3.2.3.1.3.2.2

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

$x^{4} - 8 x^{1 + 2} - 8 x \cdot x^{2} - 8 x (- 8 x) - 64 x^{2} = 0$

Step 1.2.3.2.3.1.3.3

Add $1$ and $2$ .

$x^{4} - 8 x^{3} - 8 x \cdot x^{2} - 8 x (- 8 x) - 64 x^{2} = 0$

Step 1.2.3.2.3.1.4

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

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Step 1.2.3.2.3.1.4.1

Move $x^{2}$ .

$x^{4} - 8 x^{3} - 8 (x^{2} x) - 8 x (- 8 x) - 64 x^{2} = 0$

Step 1.2.3.2.3.1.4.2

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

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Step 1.2.3.2.3.1.4.2.1

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

$x^{4} - 8 x^{3} - 8 (x^{2} x^{1}) - 8 x (- 8 x) - 64 x^{2} = 0$

Step 1.2.3.2.3.1.4.2.2

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

$x^{4} - 8 x^{3} - 8 x^{2 + 1} - 8 x (- 8 x) - 64 x^{2} = 0$

Step 1.2.3.2.3.1.4.3

Add $2$ and $1$ .

$x^{4} - 8 x^{3} - 8 x^{3} - 8 x (- 8 x) - 64 x^{2} = 0$

Step 1.2.3.2.3.1.5

Rewrite using the commutative property of multiplication.

$x^{4} - 8 x^{3} - 8 x^{3} - 8 \cdot - 8 x \cdot x - 64 x^{2} = 0$

Step 1.2.3.2.3.1.6

Multiply $x$ by $x$ by adding the exponents.

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Step 1.2.3.2.3.1.6.1

Move $x$ .

$x^{4} - 8 x^{3} - 8 x^{3} - 8 \cdot - 8 (x \cdot x) - 64 x^{2} = 0$

Step 1.2.3.2.3.1.6.2

Multiply $x$ by $x$ .

$x^{4} - 8 x^{3} - 8 x^{3} - 8 \cdot - 8 x^{2} - 64 x^{2} = 0$

Step 1.2.3.2.3.1.7

Multiply $- 8$ by $- 8$ .

$x^{4} - 8 x^{3} - 8 x^{3} + 64 x^{2} - 64 x^{2} = 0$

Step 1.2.3.2.3.2

Subtract $8 x^{3}$ from $- 8 x^{3}$ .

$x^{4} - 16 x^{3} + 64 x^{2} - 64 x^{2} = 0$

Step 1.2.3.3

Combine the opposite terms in $x^{4} - 16 x^{3} + 64 x^{2} - 64 x^{2}$ .

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Step 1.2.3.3.1

Subtract $64 x^{2}$ from $64 x^{2}$ .

$x^{4} - 16 x^{3} + 0 = 0$

Step 1.2.3.3.2

Add $x^{4} - 16 x^{3}$ and $0$ .

$x^{4} - 16 x^{3} = 0$

Step 1.2.4

Factor $x^{3}$ out of $x^{4} - 16 x^{3}$ .

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Step 1.2.4.1

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

$x^{3} x - 16 x^{3} = 0$

Step 1.2.4.2

Factor $x^{3}$ out of $- 16 x^{3}$ .

$x^{3} x + x^{3} \cdot - 16 = 0$

Step 1.2.4.3

Factor $x^{3}$ out of $x^{3} x + x^{3} \cdot - 16$ .

$x^{3} (x - 16) = 0$

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 $0$ .

$x^{3} = 0$

$x - 16 = 0$

Step 1.2.6

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

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Step 1.2.6.1

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

$x^{3} = 0$

Step 1.2.6.2

Solve $x^{3} = 0$ 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.

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

Step 1.2.6.2.2

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

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Step 1.2.6.2.2.1

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

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

Step 1.2.6.2.2.2

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

$x = 0$

Step 1.2.7

Set $x - 16$ equal to $0$ and solve for $x$ .

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Step 1.2.7.1

Set $x - 16$ equal to $0$ .

$x - 16 = 0$

Step 1.2.7.2

Add $16$ to both sides of the equation.

$x = 16$

Step 1.2.8

The final solution is all the values that make $x^{3} (x - 16) = 0$ true.

$x = 0, 16$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (16, 0)$

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 $y$ .

${({(0)}^{2} + y^{2} - 8 \cdot 0)}^{2} = 64 {(0)}^{2} + 64 y^{2}$

Step 2.2

Solve the equation.

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Step 2.2.1

Simplify ${({(0)}^{2} + y^{2} - 8 \cdot 0)}^{2}$ .

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Step 2.2.1.1

Rewrite.

$0 + 0 + {({(0)}^{2} + y^{2} - 8 \cdot 0)}^{2} = 64 {(0)}^{2} + 64 y^{2}$

Step 2.2.1.2

Simplify by adding zeros.

${({(0)}^{2} + y^{2} - 8 \cdot 0)}^{2} = 64 {(0)}^{2} + 64 y^{2}$

Step 2.2.1.3

Simplify each term.

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Step 2.2.1.3.1

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

${(0 + y^{2} - 8 \cdot 0)}^{2} = 64 {(0)}^{2} + 64 y^{2}$

Step 2.2.1.3.2

Multiply $- 8$ by $0$ .

${(0 + y^{2} + 0)}^{2} = 64 {(0)}^{2} + 64 y^{2}$

Step 2.2.1.4

Simplify by adding terms.

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Step 2.2.1.4.1

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

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Step 2.2.1.4.1.1

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

${(y^{2} + 0)}^{2} = 64 {(0)}^{2} + 64 y^{2}$

Step 2.2.1.4.1.2

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

${(y^{2})}^{2} = 64 {(0)}^{2} + 64 y^{2}$

Step 2.2.1.4.2

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

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Step 2.2.1.4.2.1

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

$y^{2 \cdot 2} = 64 {(0)}^{2} + 64 y^{2}$

Step 2.2.1.4.2.2

Multiply $2$ by $2$ .

$y^{4} = 64 {(0)}^{2} + 64 y^{2}$

Step 2.2.2

Simplify $64 {(0)}^{2} + 64 y^{2}$ .

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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 $0$ .

$y^{4} = 64 \cdot 0 + 64 y^{2}$

Step 2.2.2.1.2

Multiply $64$ by $0$ .

$y^{4} = 0 + 64 y^{2}$

Step 2.2.2.2

Add $0$ and $64 y^{2}$ .

$y^{4} = 64 y^{2}$

Step 2.2.3

Subtract $64 y^{2}$ from both sides of the equation.

$y^{4} - 64 y^{2} = 0$

Step 2.2.4

Factor the left side of the equation.

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Step 2.2.4.1

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

${(y^{2})}^{2} - 64 y^{2} = 0$

Step 2.2.4.2

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

$u^{2} - 64 u = 0$

Step 2.2.4.3

Factor $u$ out of $u^{2} - 64 u$ .

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Step 2.2.4.3.1

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

$u \cdot u - 64 u = 0$

Step 2.2.4.3.2

Factor $u$ out of $- 64 u$ .

$u \cdot u + u \cdot - 64 = 0$

Step 2.2.4.3.3

Factor $u$ out of $u \cdot u + u \cdot - 64$ .

$u (u - 64) = 0$

Step 2.2.4.4

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

$y^{2} (y^{2} - 64) = 0$

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 $0$ .

$y^{2} = 0$

$y^{2} - 64 = 0$

Step 2.2.6

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

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Step 2.2.6.1

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

$y^{2} = 0$

Step 2.2.6.2

Solve $y^{2} = 0$ 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.

$y = \pm \sqrt{0}$

Step 2.2.6.2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.2.6.2.2.1

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

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

Step 2.2.6.2.2.2

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

$y = \pm 0$

Step 2.2.6.2.2.3

Plus or minus $0$ is $0$ .

$y = 0$

Step 2.2.7

Set $y^{2} - 64$ equal to $0$ and solve for $y$ .

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Step 2.2.7.1

Set $y^{2} - 64$ equal to $0$ .

$y^{2} - 64 = 0$

Step 2.2.7.2

Solve $y^{2} - 64 = 0$ for $y$ .

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Step 2.2.7.2.1

Add $64$ to both sides of the equation.

$y^{2} = 64$

Step 2.2.7.2.2

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

$y = \pm \sqrt{64}$

Step 2.2.7.2.3

Simplify $\pm \sqrt{64}$ .

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Step 2.2.7.2.3.1

Rewrite $64$ as $8^{2}$ .

$y = \pm \sqrt{8^{2}}$

Step 2.2.7.2.3.2

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

$y = \pm 8$

Step 2.2.7.2.4

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

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Step 2.2.7.2.4.1

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

$y = 8$

Step 2.2.7.2.4.2

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

$y = - 8$

Step 2.2.7.2.4.3

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

$y = 8, - 8$

Step 2.2.8

The final solution is all the values that make $y^{2} (y^{2} - 64) = 0$ true.

$y = 0, 8, - 8$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 0), (0, 8), (0, - 8)$

Step 3

List the intersections.

x-intercept(s): $(0, 0), (16, 0)$

y-intercept(s): $(0, 0), (0, 8), (0, - 8)$

Step 4