Algebra Examples

$y = (2 x + 3) (x - 1) (x - 4)$

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 $x$ .

$0 = (2 x + 3) (x - 1) (x - 4)$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $(2 x + 3) (x - 1) (x - 4) = 0$ .

$(2 x + 3) (x - 1) (x - 4) = 0$

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 $0$ .

$2 x + 3 = 0$

$x - 1 = 0$

$x - 4 = 0$

Step 1.2.3

Set $2 x + 3$ equal to $0$ and solve for $x$ .

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Step 1.2.3.1

Set $2 x + 3$ equal to $0$ .

$2 x + 3 = 0$

Step 1.2.3.2

Solve $2 x + 3 = 0$ for $x$ .

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Step 1.2.3.2.1

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Step 1.2.3.2.2

Divide each term in $2 x = - 3$ by $2$ and simplify.

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Step 1.2.3.2.2.1

Divide each term in $2 x = - 3$ by $2$ .

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 1.2.3.2.2.2

Simplify the left side.

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Step 1.2.3.2.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.3.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 1.2.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{2}$

Step 1.2.3.2.2.3

Simplify the right side.

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Step 1.2.3.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{2}$

Step 1.2.4

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 1.2.4.2

Add $1$ to both sides of the equation.

$x = 1$

Step 1.2.5

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 1.2.5.2

Add $4$ to both sides of the equation.

$x = 4$

Step 1.2.6

The final solution is all the values that make $(2 x + 3) (x - 1) (x - 4) = 0$ true.

$x = - \frac{3}{2}, 1, 4$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- \frac{3}{2}, 0), (1, 0), (4, 0)$

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 $y$ .

$y = (2 (0) + 3) ((0) - 1) ((0) - 4)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = (2 (0) + 3) (0 - 1) ((0) - 4)$

Step 2.2.2

Remove parentheses.

$y = (2 (0) + 3) (0 - 1) (0 - 4)$

Step 2.2.3

Remove parentheses.

$y = (2 (0) + 3) ((0) - 1) ((0) - 4)$

Step 2.2.4

Simplify $(2 (0) + 3) ((0) - 1) ((0) - 4)$ .

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Step 2.2.4.1

Multiply $2$ by $0$ .

$y = (0 + 3) ((0) - 1) ((0) - 4)$

Step 2.2.4.2

Add $0$ and $3$ .

$y = 3 ((0) - 1) ((0) - 4)$

Step 2.2.4.3

Subtract $1$ from $0$ .

$y = 3 \cdot - 1 ((0) - 4)$

Step 2.2.4.4

Multiply $3$ by $- 1$ .

$y = - 3 ((0) - 4)$

Step 2.2.4.5

Subtract $4$ from $0$ .

$y = - 3 \cdot - 4$

Step 2.2.4.6

Multiply $- 3$ by $- 4$ .

$y = 12$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 12)$

Step 3

List the intersections.

x-intercept(s): $(- \frac{3}{2}, 0), (1, 0), (4, 0)$

y-intercept(s): $(0, 12)$

Step 4