Algebra Examples

$f (x) = x (x - 7) (6 x + 5) (x - 7)$

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 $x$ .

$0 = x (x - 7) (6 x + 5) (x - 7)$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $x (x - 7) (6 x + 5) (x - 7) = 0$ .

$x (x - 7) (6 x + 5) (x - 7) = 0$

Step 1.2.2

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

$x = 0$

$x - 7 = 0$

$6 x + 5 = 0$

$x - 7 = 0$

Step 1.2.3

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.4

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

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Step 1.2.4.1

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

$x - 7 = 0$

Step 1.2.4.2

Add $7$ to both sides of the equation.

$x = 7$

Step 1.2.5

Set $6 x + 5$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $6 x + 5$ equal to $0$ .

$6 x + 5 = 0$

Step 1.2.5.2

Solve $6 x + 5 = 0$ for $x$ .

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Step 1.2.5.2.1

Subtract $5$ from both sides of the equation.

$6 x = - 5$

Step 1.2.5.2.2

Divide each term in $6 x = - 5$ by $6$ and simplify.

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Step 1.2.5.2.2.1

Divide each term in $6 x = - 5$ by $6$ .

$\frac{6 x}{6} = \frac{- 5}{6}$

Step 1.2.5.2.2.2

Simplify the left side.

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Step 1.2.5.2.2.2.1

Cancel the common factor of $6$ .

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Step 1.2.5.2.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{- 5}{6}$

Step 1.2.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{6}$

Step 1.2.5.2.2.3

Simplify the right side.

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Step 1.2.5.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{5}{6}$

Step 1.2.6

The final solution is all the values that make $x (x - 7) (6 x + 5) (x - 7) = 0$ true.

$x = 0, 7, - \frac{5}{6}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (7, 0), (- \frac{5}{6}, 0)$

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 $y$ .

$y = (0) ((0) - 7) (6 (0) + 5) ((0) - 7)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 0 (0 - 7) (6 (0) + 5) ((0) - 7)$

Step 2.2.2

Remove parentheses.

$y = 0 (0 - 7) (6 (0) + 5) (0 - 7)$

Step 2.2.3

Remove parentheses.

$y = (0) ((0) - 7) (6 (0) + 5) ((0) - 7)$

Step 2.2.4

Simplify $(0) ((0) - 7) (6 (0) + 5) ((0) - 7)$ .

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Step 2.2.4.1

Subtract $7$ from $0$ .

$y = 0 \cdot - 7 (6 (0) + 5) ((0) - 7)$

Step 2.2.4.2

Multiply $0$ by $- 7$ .

$y = 0 (6 (0) + 5) ((0) - 7)$

Step 2.2.4.3

Multiply $6$ by $0$ .

$y = 0 (0 + 5) ((0) - 7)$

Step 2.2.4.4

Add $0$ and $5$ .

$y = 0 \cdot 5 ((0) - 7)$

Step 2.2.4.5

Multiply $0$ by $5$ .

$y = 0 ((0) - 7)$

Step 2.2.4.6

Subtract $7$ from $0$ .

$y = 0 \cdot - 7$

Step 2.2.4.7

Multiply $0$ by $- 7$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (7, 0), (- \frac{5}{6}, 0)$

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

Step 4