Finite Math Examples

$\sqrt{x} + \sqrt{y} = \sqrt{45}$

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 $x$ .

$\sqrt{x} + \sqrt{0} = \sqrt{45}$

Step 1.2

Solve the equation.

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Step 1.2.1

Solve for $\sqrt{x}$ .

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Step 1.2.1.1

Simplify $\sqrt{x} + \sqrt{0}$ .

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Step 1.2.1.1.1

Simplify each term.

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Step 1.2.1.1.1.1

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

$\sqrt{x} + \sqrt{0^{2}} = \sqrt{45}$

Step 1.2.1.1.1.2

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

$\sqrt{x} + 0 = \sqrt{45}$

Step 1.2.1.1.2

Add $\sqrt{x}$ and $0$ .

$\sqrt{x} = \sqrt{45}$

Step 1.2.1.2

Simplify $\sqrt{45}$ .

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Step 1.2.1.2.1

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 1.2.1.2.1.1

Factor $9$ out of $45$ .

$\sqrt{x} = \sqrt{9 (5)}$

Step 1.2.1.2.1.2

Rewrite $9$ as $3^{2}$ .

$\sqrt{x} = \sqrt{3^{2} \cdot 5}$

Step 1.2.1.2.2

Pull terms out from under the radical.

$\sqrt{x} = 3 \sqrt{5}$

Step 1.2.2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x}}^{2} = {(3 \sqrt{5})}^{2}$

Step 1.2.3

Simplify each side of the equation.

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Step 1.2.3.1

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

${(x^{\frac{1}{2}})}^{2} = {(3 \sqrt{5})}^{2}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

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

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Step 1.2.3.2.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 1.2.3.2.1.1.1

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

$x^{\frac{1}{2} \cdot 2} = {(3 \sqrt{5})}^{2}$

Step 1.2.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 1.2.3.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(3 \sqrt{5})}^{2}$

Step 1.2.3.2.1.1.2.2

Rewrite the expression.

$x^{1} = {(3 \sqrt{5})}^{2}$

Step 1.2.3.2.1.2

Simplify.

$x = {(3 \sqrt{5})}^{2}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Simplify ${(3 \sqrt{5})}^{2}$ .

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Step 1.2.3.3.1.1

Simplify the expression.

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Step 1.2.3.3.1.1.1

Apply the product rule to $3 \sqrt{5}$ .

$x = 3^{2} {\sqrt{5}}^{2}$

Step 1.2.3.3.1.1.2

Raise $3$ to the power of $2$ .

$x = 9 {\sqrt{5}}^{2}$

Step 1.2.3.3.1.2

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 1.2.3.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$x = 9 {(5^{\frac{1}{2}})}^{2}$

Step 1.2.3.3.1.2.2

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

$x = 9 \cdot 5^{\frac{1}{2} \cdot 2}$

Step 1.2.3.3.1.2.3

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

$x = 9 \cdot 5^{\frac{2}{2}}$

Step 1.2.3.3.1.2.4

Cancel the common factor of $2$ .

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Step 1.2.3.3.1.2.4.1

Cancel the common factor.

$x = 9 \cdot 5^{\frac{2}{2}}$

Step 1.2.3.3.1.2.4.2

Rewrite the expression.

$x = 9 \cdot 5^{1}$

Step 1.2.3.3.1.2.5

Evaluate the exponent.

$x = 9 \cdot 5$

Step 1.2.3.3.1.3

Multiply $9$ by $5$ .

$x = 45$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(45, 0)$

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 $y$ .

$\sqrt{0} + \sqrt{y} = \sqrt{45}$

Step 2.2

Solve the equation.

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Step 2.2.1

Solve for $\sqrt{y}$ .

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Step 2.2.1.1

Simplify $\sqrt{0} + \sqrt{y}$ .

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Step 2.2.1.1.1

Simplify each term.

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Step 2.2.1.1.1.1

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

$\sqrt{0^{2}} + \sqrt{y} = \sqrt{45}$

Step 2.2.1.1.1.2

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

$0 + \sqrt{y} = \sqrt{45}$

Step 2.2.1.1.2

Add $0$ and $\sqrt{y}$ .

$\sqrt{y} = \sqrt{45}$

Step 2.2.1.2

Simplify $\sqrt{45}$ .

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Step 2.2.1.2.1

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 2.2.1.2.1.1

Factor $9$ out of $45$ .

$\sqrt{y} = \sqrt{9 (5)}$

Step 2.2.1.2.1.2

Rewrite $9$ as $3^{2}$ .

$\sqrt{y} = \sqrt{3^{2} \cdot 5}$

Step 2.2.1.2.2

Pull terms out from under the radical.

$\sqrt{y} = 3 \sqrt{5}$

Step 2.2.2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{y}}^{2} = {(3 \sqrt{5})}^{2}$

Step 2.2.3

Simplify each side of the equation.

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Step 2.2.3.1

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

${(y^{\frac{1}{2}})}^{2} = {(3 \sqrt{5})}^{2}$

Step 2.2.3.2

Simplify the left side.

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Step 2.2.3.2.1

Simplify ${(y^{\frac{1}{2}})}^{2}$ .

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Step 2.2.3.2.1.1

Multiply the exponents in ${(y^{\frac{1}{2}})}^{2}$ .

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Step 2.2.3.2.1.1.1

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

$y^{\frac{1}{2} \cdot 2} = {(3 \sqrt{5})}^{2}$

Step 2.2.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.3.2.1.1.2.1

Cancel the common factor.

$y^{\frac{1}{2} \cdot 2} = {(3 \sqrt{5})}^{2}$

Step 2.2.3.2.1.1.2.2

Rewrite the expression.

$y^{1} = {(3 \sqrt{5})}^{2}$

Step 2.2.3.2.1.2

Simplify.

$y = {(3 \sqrt{5})}^{2}$

Step 2.2.3.3

Simplify the right side.

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Step 2.2.3.3.1

Simplify ${(3 \sqrt{5})}^{2}$ .

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Step 2.2.3.3.1.1

Simplify the expression.

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Step 2.2.3.3.1.1.1

Apply the product rule to $3 \sqrt{5}$ .

$y = 3^{2} {\sqrt{5}}^{2}$

Step 2.2.3.3.1.1.2

Raise $3$ to the power of $2$ .

$y = 9 {\sqrt{5}}^{2}$

Step 2.2.3.3.1.2

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 2.2.3.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$y = 9 {(5^{\frac{1}{2}})}^{2}$

Step 2.2.3.3.1.2.2

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

$y = 9 \cdot 5^{\frac{1}{2} \cdot 2}$

Step 2.2.3.3.1.2.3

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

$y = 9 \cdot 5^{\frac{2}{2}}$

Step 2.2.3.3.1.2.4

Cancel the common factor of $2$ .

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Step 2.2.3.3.1.2.4.1

Cancel the common factor.

$y = 9 \cdot 5^{\frac{2}{2}}$

Step 2.2.3.3.1.2.4.2

Rewrite the expression.

$y = 9 \cdot 5^{1}$

Step 2.2.3.3.1.2.5

Evaluate the exponent.

$y = 9 \cdot 5$

Step 2.2.3.3.1.3

Multiply $9$ by $5$ .

$y = 45$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 45)$

Step 3

List the intersections.

x-intercept(s): $(45, 0)$

y-intercept(s): $(0, 45)$

Step 4