Algebra Examples

$y = - 2 x^{2} + 9 x + 18$

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 $x$ .

$0 = - 2 x^{2} + 9 x + 18$

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Rewrite the equation as $- 2 x^{2} + 9 x + 18 = 0$ .

$- 2 x^{2} + 9 x + 18 = 0$

Step 1.2.2

Factor the left side of the equation.

Tap for more steps...

Step 1.2.2.1

Factor $- 1$ out of $- 2 x^{2} + 9 x + 18$ .

Tap for more steps...

Step 1.2.2.1.1

Factor $- 1$ out of $- 2 x^{2}$ .

$- (2 x^{2}) + 9 x + 18 = 0$

Step 1.2.2.1.2

Factor $- 1$ out of $9 x$ .

$- (2 x^{2}) - (- 9 x) + 18 = 0$

Step 1.2.2.1.3

Rewrite $18$ as $- 1 (- 18)$ .

$- (2 x^{2}) - (- 9 x) - 1 \cdot - 18 = 0$

Step 1.2.2.1.4

Factor $- 1$ out of $- (2 x^{2}) - (- 9 x)$ .

$- (2 x^{2} - 9 x) - 1 \cdot - 18 = 0$

Step 1.2.2.1.5

Factor $- 1$ out of $- (2 x^{2} - 9 x) - 1 (- 18)$ .

$- (2 x^{2} - 9 x - 18) = 0$

Step 1.2.2.2

Factor.

Tap for more steps...

Step 1.2.2.2.1

Factor by grouping.

Tap for more steps...

Step 1.2.2.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 18 = - 36$ and whose sum is $b = - 9$ .

Tap for more steps...

Step 1.2.2.2.1.1.1

Factor $- 9$ out of $- 9 x$ .

$- (2 x^{2} - 9 x - 18) = 0$

Step 1.2.2.2.1.1.2

Rewrite $- 9$ as $3$ plus $- 12$

$- (2 x^{2} + (3 - 12) x - 18) = 0$

Step 1.2.2.2.1.1.3

Apply the distributive property.

$- (2 x^{2} + 3 x - 12 x - 18) = 0$

Step 1.2.2.2.1.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.2.2.2.1.2.1

Group the first two terms and the last two terms.

$- ((2 x^{2} + 3 x) - 12 x - 18) = 0$

Step 1.2.2.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$- (x (2 x + 3) - 6 (2 x + 3)) = 0$

Step 1.2.2.2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 3$ .

$- ((2 x + 3) (x - 6)) = 0$

Step 1.2.2.2.2

Remove unnecessary parentheses.

$- (2 x + 3) (x - 6) = 0$

Step 1.2.3

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

$2 x + 3 = 0$

$x - 6 = 0$

Step 1.2.4

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

Tap for more steps...

Step 1.2.4.1

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

$2 x + 3 = 0$

Step 1.2.4.2

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

Tap for more steps...

Step 1.2.4.2.1

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Step 1.2.4.2.2

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

Tap for more steps...

Step 1.2.4.2.2.1

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

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

Step 1.2.4.2.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.4.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.4.2.2.2.1.1

Cancel the common factor.

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

Step 1.2.4.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 1.2.4.2.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.4.2.2.3.1

Move the negative in front of the fraction.

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

Step 1.2.5

Set $x - 6$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.5.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 1.2.5.2

Add $6$ to both sides of the equation.

$x = 6$

Step 1.2.6

The final solution is all the values that make $- (2 x + 3) (x - 6) = 0$ true.

$x = - \frac{3}{2}, 6$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- \frac{3}{2}, 0), (6, 0)$

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 $y$ .

$y = - 2 {(0)}^{2} + 9 (0) + 18$

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Remove parentheses.

$y = - 2 \cdot 0^{2} + 9 (0) + 18$

Step 2.2.2

Remove parentheses.

$y = - 2 {(0)}^{2} + 9 (0) + 18$

Step 2.2.3

Simplify $- 2 {(0)}^{2} + 9 (0) + 18$ .

Tap for more steps...

Step 2.2.3.1

Simplify each term.

Tap for more steps...

Step 2.2.3.1.1

Raising $0$ to any positive power yields $0$ .

$y = - 2 \cdot 0 + 9 (0) + 18$

Step 2.2.3.1.2

Multiply $- 2$ by $0$ .

$y = 0 + 9 (0) + 18$

Step 2.2.3.1.3

Multiply $9$ by $0$ .

$y = 0 + 0 + 18$

Step 2.2.3.2

Simplify by adding numbers.

Tap for more steps...

Step 2.2.3.2.1

Add $0$ and $0$ .

$y = 0 + 18$

Step 2.2.3.2.2

Add $0$ and $18$ .

$y = 18$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 18)$

Step 3

List the intersections.

x-intercept(s): $(- \frac{3}{2}, 0), (6, 0)$

y-intercept(s): $(0, 18)$

Step 4