Algebra Examples

$y = \frac{2 - \sqrt{x}}{5 x + 1}$

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 $x$ .

$0 = \frac{2 - \sqrt{x}}{5 x + 1}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$2 - \sqrt{x} = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Subtract $2$ from both sides of the equation.

$- \sqrt{x} = - 2$

Step 1.2.2.2

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

${(- \sqrt{x})}^{2} = {(- 2)}^{2}$

Step 1.2.2.3

Simplify each side of the equation.

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Step 1.2.2.3.1

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

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

Step 1.2.2.3.2

Simplify the left side.

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Step 1.2.2.3.2.1

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

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Step 1.2.2.3.2.1.1

Apply the product rule to $- x^{\frac{1}{2}}$ .

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

Step 1.2.2.3.2.1.2

Raise $- 1$ to the power of $2$ .

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

Step 1.2.2.3.2.1.3

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

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

Step 1.2.2.3.2.1.4

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

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Step 1.2.2.3.2.1.4.1

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

$x^{\frac{1}{2} \cdot 2} = {(- 2)}^{2}$

Step 1.2.2.3.2.1.4.2

Cancel the common factor of $2$ .

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Step 1.2.2.3.2.1.4.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(- 2)}^{2}$

Step 1.2.2.3.2.1.4.2.2

Rewrite the expression.

$x^{1} = {(- 2)}^{2}$

Step 1.2.2.3.2.1.5

Simplify.

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

Step 1.2.2.3.3

Simplify the right side.

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Step 1.2.2.3.3.1

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

$x = 4$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(4, 0)$

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 $y$ .

$y = \frac{2 - \sqrt{0}}{5 (0) + 1}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{2 - \sqrt{0}}{5 (0) + 1}$

Step 2.2.2

Simplify $\frac{2 - \sqrt{0}}{5 (0) + 1}$ .

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Step 2.2.2.1

Simplify the numerator.

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Step 2.2.2.1.1

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

$y = \frac{2 - \sqrt{0^{2}}}{5 (0) + 1}$

Step 2.2.2.1.2

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

$y = \frac{2 - 0}{5 (0) + 1}$

Step 2.2.2.1.3

Multiply $- 1$ by $0$ .

$y = \frac{2 + 0}{5 (0) + 1}$

Step 2.2.2.1.4

Add $2$ and $0$ .

$y = \frac{2}{5 (0) + 1}$

Step 2.2.2.2

Simplify the denominator.

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Step 2.2.2.2.1

Multiply $5$ by $0$ .

$y = \frac{2}{0 + 1}$

Step 2.2.2.2.2

Add $0$ and $1$ .

$y = \frac{2}{1}$

Step 2.2.2.3

Divide $2$ by $1$ .

$y = 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 2)$

Step 3

List the intersections.

x-intercept(s): $(4, 0)$

y-intercept(s): $(0, 2)$

Step 4