Find the x and y Intercepts f(x)=-4x^2-6x+1

$f (x) = - 4 x^{2} - 6 x + 1$

Step 1

Find the x-intercepts.

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

$0 = - 4 x^{2} - 6 x + 1$

Solve the equation.

Rewrite the equation as $- 4 x^{2} - 6 x + 1 = 0$ .

$- 4 x^{2} - 6 x + 1 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = - 4$ , $b = - 6$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (- 4 \cdot 1)}}{2 \cdot - 4}$

Simplify.

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 4 \cdot 1}}{2 \cdot - 4}$

Multiply $- 4 \cdot - 4 \cdot 1$ .

Multiply $- 4$ by $- 4$ .

$x = \frac{6 \pm \sqrt{36 + 16 \cdot 1}}{2 \cdot - 4}$

Multiply $16$ by $1$ .

$x = \frac{6 \pm \sqrt{36 + 16}}{2 \cdot - 4}$

Add $36$ and $16$ .

$x = \frac{6 \pm \sqrt{52}}{2 \cdot - 4}$

Rewrite $52$ as $2^{2} \cdot 13$ .

Factor $4$ out of $52$ .

$x = \frac{6 \pm \sqrt{4 (13)}}{2 \cdot - 4}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{6 \pm \sqrt{2^{2} \cdot 13}}{2 \cdot - 4}$

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{13}}{2 \cdot - 4}$

Multiply $2$ by $- 4$ .

$x = \frac{6 \pm 2 \sqrt{13}}{- 8}$

Simplify $\frac{6 \pm 2 \sqrt{13}}{- 8}$ .

$x = \frac{3 \pm \sqrt{13}}{- 4}$

Move the negative in front of the fraction.

$x = - \frac{3 \pm \sqrt{13}}{4}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 4 \cdot 1}}{2 \cdot - 4}$

Multiply $- 4 \cdot - 4 \cdot 1$ .

Multiply $- 4$ by $- 4$ .

$x = \frac{6 \pm \sqrt{36 + 16 \cdot 1}}{2 \cdot - 4}$

Multiply $16$ by $1$ .

$x = \frac{6 \pm \sqrt{36 + 16}}{2 \cdot - 4}$

Add $36$ and $16$ .

$x = \frac{6 \pm \sqrt{52}}{2 \cdot - 4}$

Rewrite $52$ as $2^{2} \cdot 13$ .

Factor $4$ out of $52$ .

$x = \frac{6 \pm \sqrt{4 (13)}}{2 \cdot - 4}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{6 \pm \sqrt{2^{2} \cdot 13}}{2 \cdot - 4}$

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{13}}{2 \cdot - 4}$

Multiply $2$ by $- 4$ .

$x = \frac{6 \pm 2 \sqrt{13}}{- 8}$

Simplify $\frac{6 \pm 2 \sqrt{13}}{- 8}$ .

$x = \frac{3 \pm \sqrt{13}}{- 4}$

Move the negative in front of the fraction.

$x = - \frac{3 \pm \sqrt{13}}{4}$

Change the $\pm$ to $+$ .

$x = - \frac{3 + \sqrt{13}}{4}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 4 \cdot 1}}{2 \cdot - 4}$

Multiply $- 4 \cdot - 4 \cdot 1$ .

Multiply $- 4$ by $- 4$ .

$x = \frac{6 \pm \sqrt{36 + 16 \cdot 1}}{2 \cdot - 4}$

Multiply $16$ by $1$ .

$x = \frac{6 \pm \sqrt{36 + 16}}{2 \cdot - 4}$

Add $36$ and $16$ .

$x = \frac{6 \pm \sqrt{52}}{2 \cdot - 4}$

Rewrite $52$ as $2^{2} \cdot 13$ .

Factor $4$ out of $52$ .

$x = \frac{6 \pm \sqrt{4 (13)}}{2 \cdot - 4}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{6 \pm \sqrt{2^{2} \cdot 13}}{2 \cdot - 4}$

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{13}}{2 \cdot - 4}$

Multiply $2$ by $- 4$ .

$x = \frac{6 \pm 2 \sqrt{13}}{- 8}$

Simplify $\frac{6 \pm 2 \sqrt{13}}{- 8}$ .

$x = \frac{3 \pm \sqrt{13}}{- 4}$

Move the negative in front of the fraction.

$x = - \frac{3 \pm \sqrt{13}}{4}$

Change the $\pm$ to $-$ .

$x = - \frac{3 - \sqrt{13}}{4}$

The final answer is the combination of both solutions.

$x = - \frac{3 + \sqrt{13}}{4}, - \frac{3 - \sqrt{13}}{4}$

x-intercept(s) in point form.

x-intercept(s): $(- \frac{3 + \sqrt{13}}{4}, 0), (- \frac{3 - \sqrt{13}}{4}, 0)$

Step 2

Find the y-intercepts.

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

$y = - 4 {(0)}^{2} - 6 \cdot 0 + 1$

Solve the equation.

Remove parentheses.

$y = - 4 \cdot 0^{2} - 6 \cdot 0 + 1$

Remove parentheses.

$y = - 4 {(0)}^{2} - 6 \cdot 0 + 1$

Simplify $- 4 {(0)}^{2} - 6 \cdot 0 + 1$ .

Simplify each term.

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

$y = - 4 \cdot 0 - 6 \cdot 0 + 1$

Multiply $- 4$ by $0$ .

$y = 0 - 6 \cdot 0 + 1$

Multiply $- 6$ by $0$ .

$y = 0 + 0 + 1$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 1$

Add $0$ and $1$ .

$y = 1$

y-intercept(s) in point form.

y-intercept(s): $(0, 1)$

Step 3

List the intersections.

x-intercept(s): $(- \frac{3 + \sqrt{13}}{4}, 0), (- \frac{3 - \sqrt{13}}{4}, 0)$

y-intercept(s): $(0, 1)$

Step 4

Find the x and y Intercepts f(x)=-4x^2-6x+1

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