Algebra Examples

$f (x) = 2 (2 x + 3) {(x - 5)}^{2}$

Step 1

Find the x-intercepts.

Tap for more steps...

Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = 2 (2 x + 3) {(x - 5)}^{2}$

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Rewrite the equation as $2 (2 x + 3) {(x - 5)}^{2} = 0$ .

$2 (2 x + 3) {(x - 5)}^{2} = 0$

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 $0$ .

$2 x + 3 = 0$

${(x - 5)}^{2} = 0$

Step 1.2.3

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

Tap for more steps...

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 $x$ .

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

Step 1.2.3.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

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.

Tap for more steps...

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 ${(x - 5)}^{2}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.4.1

Set ${(x - 5)}^{2}$ equal to $0$ .

${(x - 5)}^{2} = 0$

Step 1.2.4.2

Solve ${(x - 5)}^{2} = 0$ for $x$ .

Tap for more steps...

Step 1.2.4.2.1

Set the $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 1.2.4.2.2

Add $5$ to both sides of the equation.

$x = 5$

Step 1.2.5

The final solution is all the values that make $2 (2 x + 3) {(x - 5)}^{2} = 0$ true.

$x = - \frac{3}{2}, 5$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- \frac{3}{2}, 0), (5, 0)$

Step 2

Find the y-intercepts.

Tap for more steps...

Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = 2 (2 (0) + 3) {((0) - 5)}^{2}$

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Remove parentheses.

$y = 2 (2 (0) + 3) {(0 - 5)}^{2}$

Step 2.2.2

Remove parentheses.

$y = 2 (2 (0) + 3) {((0) - 5)}^{2}$

Step 2.2.3

Simplify $2 (2 (0) + 3) {((0) - 5)}^{2}$ .

Tap for more steps...

Step 2.2.3.1

Multiply $2$ by $0$ .

$y = 2 (0 + 3) {((0) - 5)}^{2}$

Step 2.2.3.2

Add $0$ and $3$ .

$y = 2 \cdot 3 {((0) - 5)}^{2}$

Step 2.2.3.3

Multiply $2$ by $3$ .

$y = 6 {((0) - 5)}^{2}$

Step 2.2.3.4

Subtract $5$ from $0$ .

$y = 6 {(- 5)}^{2}$

Step 2.2.3.5

Raise $- 5$ to the power of $2$ .

$y = 6 \cdot 25$

Step 2.2.3.6

Multiply $6$ by $25$ .

$y = 150$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 150)$

Step 3

List the intersections.

x-intercept(s): $(- \frac{3}{2}, 0), (5, 0)$

y-intercept(s): $(0, 150)$

Step 4