Find the x and y Intercepts f(x)=4x^2-13x+10

$f (x) = 4 x^{2} - 13 x + 10$

Step 1

Find the x-intercepts.

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

$0 = 4 x^{2} - 13 x + 10$

Solve the equation.

Rewrite the equation as $4 x^{2} - 13 x + 10 = 0$ .

$4 x^{2} - 13 x + 10 = 0$

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 $a \cdot c = 4 \cdot 10 = 40$ and whose sum is $b = - 13$ .

Factor $- 13$ out of $- 13 x$ .

$4 x^{2} - 13 x + 10 = 0$

Rewrite $- 13$ as $- 5$ plus $- 8$

$4 x^{2} + (- 5 - 8) x + 10 = 0$

Apply the distributive property.

$4 x^{2} - 5 x - 8 x + 10 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(4 x^{2} - 5 x) - 8 x + 10 = 0$

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

$x (4 x - 5) - 2 (4 x - 5) = 0$

Factor the polynomial by factoring out the greatest common factor, $4 x - 5$ .

$(4 x - 5) (x - 2) = 0$

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

$4 x - 5 = 0$

$x - 2 = 0$

Set $4 x - 5$ equal to $0$ and solve for $x$ .

Set $4 x - 5$ equal to $0$ .

$4 x - 5 = 0$

Solve $4 x - 5 = 0$ for $x$ .

Add $5$ to both sides of the equation.

$4 x = 5$

Divide each term in $4 x = 5$ by $4$ and simplify.

Divide each term in $4 x = 5$ by $4$ .

$\frac{4 x}{4} = \frac{5}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{5}{4}$

Divide $x$ by $1$ .

$x = \frac{5}{4}$

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

The final solution is all the values that make $(4 x - 5) (x - 2) = 0$ true.

$x = \frac{5}{4}, 2$

x-intercept(s) in point form.

x-intercept(s): $(\frac{5}{4}, 0), (2, 0)$

Step 2

Find the y-intercepts.

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

$y = 4 {(0)}^{2} - 13 \cdot 0 + 10$

Solve the equation.

Remove parentheses.

$y = 4 \cdot 0^{2} - 13 \cdot 0 + 10$

Remove parentheses.

$y = 4 {(0)}^{2} - 13 \cdot 0 + 10$

Simplify $4 {(0)}^{2} - 13 \cdot 0 + 10$ .

Simplify each term.

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

$y = 4 \cdot 0 - 13 \cdot 0 + 10$

Multiply $4$ by $0$ .

$y = 0 - 13 \cdot 0 + 10$

Multiply $- 13$ by $0$ .

$y = 0 + 0 + 10$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 10$

Add $0$ and $10$ .

$y = 10$

y-intercept(s) in point form.

y-intercept(s): $(0, 10)$

Step 3

List the intersections.

x-intercept(s): $(\frac{5}{4}, 0), (2, 0)$

y-intercept(s): $(0, 10)$

Step 4

Find the x and y Intercepts f(x)=4x^2-13x+10

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