Find the x and y Intercepts f(x)=2x^2-15x+18

$f (x) = 2 x^{2} - 15 x + 18$

Step 1

Find the x-intercepts.

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

$0 = 2 x^{2} - 15 x + 18$

Solve the equation.

Rewrite the equation as $2 x^{2} - 15 x + 18 = 0$ .

$2 x^{2} - 15 x + 18 = 0$

Factor by grouping.

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 $b = - 15$ .

Factor $- 15$ out of $- 15 x$ .

$2 x^{2} - 15 x + 18 = 0$

Rewrite $- 15$ as $- 3$ plus $- 12$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

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

$2 x - 3 = 0$

$x - 6 = 0$

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

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

$2 x - 3 = 0$

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

Add $3$ to both sides of the equation.

$2 x = 3$

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

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

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

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

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

x-intercept(s) in point form.

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

Step 2

Find the y-intercepts.

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

$y = 2 {(0)}^{2} - 15 \cdot 0 + 18$

Solve the equation.

Remove parentheses.

$y = 2 \cdot 0^{2} - 15 \cdot 0 + 18$

Remove parentheses.

$y = 2 {(0)}^{2} - 15 \cdot 0 + 18$

Simplify $2 {(0)}^{2} - 15 \cdot 0 + 18$ .

Simplify each term.

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

$y = 2 \cdot 0 - 15 \cdot 0 + 18$

Multiply $2$ by $0$ .

$y = 0 - 15 \cdot 0 + 18$

Multiply $- 15$ by $0$ .

$y = 0 + 0 + 18$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 18$

Add $0$ and $18$ .

$y = 18$

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

Find the x and y Intercepts f(x)=2x^2-15x+18

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