Find the x and y Intercepts f(x)=3x^2-5x+2

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

Step 1

Find the x-intercepts.

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

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

Solve the equation.

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

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

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 $a \cdot c = 3 \cdot 2 = 6$ and whose sum is $b = - 5$ .

Factor $- 5$ out of $- 5 x$ .

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

Rewrite $- 5$ as $- 2$ plus $- 3$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

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

$3 x - 2 = 0$

$x - 1 = 0$

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

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

$3 x - 2 = 0$

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

Add $2$ to both sides of the equation.

$3 x = 2$

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

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

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

x-intercept(s) in point form.

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

Step 2

Find the y-intercepts.

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

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

Solve the equation.

Remove parentheses.

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

Remove parentheses.

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

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

Simplify each term.

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

$y = 3 \cdot 0 - 5 \cdot 0 + 2$

Multiply $3$ by $0$ .

$y = 0 - 5 \cdot 0 + 2$

Multiply $- 5$ by $0$ .

$y = 0 + 0 + 2$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 2$

Add $0$ and $2$ .

$y = 2$

y-intercept(s) in point form.

y-intercept(s): $(0, 2)$

Step 3

List the intersections.

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

y-intercept(s): $(0, 2)$

Step 4

Find the x and y Intercepts f(x)=3x^2-5x+2

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