Algebra Examples

$y = \frac{(2 x + 1) (x + 7)}{(x + 3) (x + 4)}$

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 $x$ .

$0 = \frac{(2 x + 1) (x + 7)}{(x + 3) (x + 4)}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$(2 x + 1) (x + 7) = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

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

$2 x + 1 = 0$

$x + 7 = 0$

Step 1.2.2.2

Set $2 x + 1$ equal to $0$ and solve for $x$ .

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Step 1.2.2.2.1

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Step 1.2.2.2.2

Solve $2 x + 1 = 0$ for $x$ .

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Step 1.2.2.2.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 1.2.2.2.2.2

Divide each term in $2 x = - 1$ by $2$ and simplify.

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Step 1.2.2.2.2.2.1

Divide each term in $2 x = - 1$ by $2$ .

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 1.2.2.2.2.2.2

Simplify the left side.

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Step 1.2.2.2.2.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.2.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 1.2.2.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{2}$

Step 1.2.2.2.2.2.3

Simplify the right side.

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Step 1.2.2.2.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

Step 1.2.2.3

Set $x + 7$ equal to $0$ and solve for $x$ .

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Step 1.2.2.3.1

Set $x + 7$ equal to $0$ .

$x + 7 = 0$

Step 1.2.2.3.2

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 1.2.2.4

The final solution is all the values that make $(2 x + 1) (x + 7) = 0$ true.

$x = - \frac{1}{2}, - 7$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- \frac{1}{2}, 0), (- 7, 0)$

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 $y$ .

$y = \frac{(2 (0) + 1) ((0) + 7)}{((0) + 3) ((0) + 4)}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{(2 (0) + 1) (0 + 7)}{((0) + 3) ((0) + 4)}$

Step 2.2.2

Remove parentheses.

$y = \frac{(2 (0) + 1) (0 + 7)}{(0 + 3) ((0) + 4)}$

Step 2.2.3

Remove parentheses.

$y = \frac{(2 (0) + 1) (0 + 7)}{(0 + 3) (0 + 4)}$

Step 2.2.4

Remove parentheses.

$y = \frac{(2 (0) + 1) ((0) + 7)}{((0) + 3) ((0) + 4)}$

Step 2.2.5

Simplify $\frac{(2 (0) + 1) ((0) + 7)}{((0) + 3) ((0) + 4)}$ .

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Step 2.2.5.1

Simplify the numerator.

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Step 2.2.5.1.1

Multiply $2$ by $0$ .

$y = \frac{(0 + 1) (0 + 7)}{(0 + 3) (0 + 4)}$

Step 2.2.5.1.2

Add $0$ and $1$ .

$y = \frac{1 (0 + 7)}{(0 + 3) (0 + 4)}$

Step 2.2.5.1.3

Multiply $0 + 7$ by $1$ .

$y = \frac{0 + 7}{(0 + 3) (0 + 4)}$

Step 2.2.5.1.4

Add $0$ and $7$ .

$y = \frac{7}{(0 + 3) (0 + 4)}$

Step 2.2.5.2

Simplify the denominator.

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Step 2.2.5.2.1

Add $0$ and $3$ .

$y = \frac{7}{3 (0 + 4)}$

Step 2.2.5.2.2

Add $0$ and $4$ .

$y = \frac{7}{3 \cdot 4}$

Step 2.2.5.3

Multiply $3$ by $4$ .

$y = \frac{7}{12}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{7}{12})$

Step 3

List the intersections.

x-intercept(s): $(- \frac{1}{2}, 0), (- 7, 0)$

y-intercept(s): $(0, \frac{7}{12})$

Step 4