Find the x and y Intercepts y=-3x^2+12x-5

$y = - 3 x^{2} + 12 x - 5$

Step 1

Find the x-intercepts.

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

$0 = - 3 x^{2} + 12 x - 5$

Solve the equation.

Rewrite the equation as $- 3 x^{2} + 12 x - 5 = 0$ .

$- 3 x^{2} + 12 x - 5 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = - 3$ , $b = 12$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

$\frac{- 12 \pm \sqrt{12^{2} - 4 \cdot (- 3 \cdot - 5)}}{2 \cdot - 3}$

Simplify.

Simplify the numerator.

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot - 3 \cdot - 5}}{2 \cdot - 3}$

Multiply $- 4 \cdot - 3 \cdot - 5$ .

Multiply $- 4$ by $- 3$ .

$x = \frac{- 12 \pm \sqrt{144 + 12 \cdot - 5}}{2 \cdot - 3}$

Multiply $12$ by $- 5$ .

$x = \frac{- 12 \pm \sqrt{144 - 60}}{2 \cdot - 3}$

Subtract $60$ from $144$ .

$x = \frac{- 12 \pm \sqrt{84}}{2 \cdot - 3}$

Rewrite $84$ as $2^{2} \cdot 21$ .

Factor $4$ out of $84$ .

$x = \frac{- 12 \pm \sqrt{4 (21)}}{2 \cdot - 3}$

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

$x = \frac{- 12 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot - 3}$

Pull terms out from under the radical.

$x = \frac{- 12 \pm 2 \sqrt{21}}{2 \cdot - 3}$

Multiply $2$ by $- 3$ .

$x = \frac{- 12 \pm 2 \sqrt{21}}{- 6}$

Simplify $\frac{- 12 \pm 2 \sqrt{21}}{- 6}$ .

$x = \frac{6 \pm \sqrt{21}}{3}$

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

Simplify the numerator.

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot - 3 \cdot - 5}}{2 \cdot - 3}$

Multiply $- 4 \cdot - 3 \cdot - 5$ .

Multiply $- 4$ by $- 3$ .

$x = \frac{- 12 \pm \sqrt{144 + 12 \cdot - 5}}{2 \cdot - 3}$

Multiply $12$ by $- 5$ .

$x = \frac{- 12 \pm \sqrt{144 - 60}}{2 \cdot - 3}$

Subtract $60$ from $144$ .

$x = \frac{- 12 \pm \sqrt{84}}{2 \cdot - 3}$

Rewrite $84$ as $2^{2} \cdot 21$ .

Factor $4$ out of $84$ .

$x = \frac{- 12 \pm \sqrt{4 (21)}}{2 \cdot - 3}$

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

$x = \frac{- 12 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot - 3}$

Pull terms out from under the radical.

$x = \frac{- 12 \pm 2 \sqrt{21}}{2 \cdot - 3}$

Multiply $2$ by $- 3$ .

$x = \frac{- 12 \pm 2 \sqrt{21}}{- 6}$

Simplify $\frac{- 12 \pm 2 \sqrt{21}}{- 6}$ .

$x = \frac{6 \pm \sqrt{21}}{3}$

Change the $\pm$ to $+$ .

$x = \frac{6 + \sqrt{21}}{3}$

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

Simplify the numerator.

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot - 3 \cdot - 5}}{2 \cdot - 3}$

Multiply $- 4 \cdot - 3 \cdot - 5$ .

Multiply $- 4$ by $- 3$ .

$x = \frac{- 12 \pm \sqrt{144 + 12 \cdot - 5}}{2 \cdot - 3}$

Multiply $12$ by $- 5$ .

$x = \frac{- 12 \pm \sqrt{144 - 60}}{2 \cdot - 3}$

Subtract $60$ from $144$ .

$x = \frac{- 12 \pm \sqrt{84}}{2 \cdot - 3}$

Rewrite $84$ as $2^{2} \cdot 21$ .

Factor $4$ out of $84$ .

$x = \frac{- 12 \pm \sqrt{4 (21)}}{2 \cdot - 3}$

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

$x = \frac{- 12 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot - 3}$

Pull terms out from under the radical.

$x = \frac{- 12 \pm 2 \sqrt{21}}{2 \cdot - 3}$

Multiply $2$ by $- 3$ .

$x = \frac{- 12 \pm 2 \sqrt{21}}{- 6}$

Simplify $\frac{- 12 \pm 2 \sqrt{21}}{- 6}$ .

$x = \frac{6 \pm \sqrt{21}}{3}$

Change the $\pm$ to $-$ .

$x = \frac{6 - \sqrt{21}}{3}$

The final answer is the combination of both solutions.

$x = \frac{6 + \sqrt{21}}{3}, \frac{6 - \sqrt{21}}{3}$

x-intercept(s) in point form.

x-intercept(s): $(\frac{6 + \sqrt{21}}{3}, 0), (\frac{6 - \sqrt{21}}{3}, 0)$

Step 2

Find the y-intercepts.

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

$y = - 3 {(0)}^{2} + 12 (0) - 5$

Solve the equation.

Remove parentheses.

$y = - 3 \cdot 0^{2} + 12 (0) - 5$

Remove parentheses.

$y = - 3 {(0)}^{2} + 12 (0) - 5$

Simplify $- 3 {(0)}^{2} + 12 (0) - 5$ .

Simplify each term.

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

$y = - 3 \cdot 0 + 12 (0) - 5$

Multiply $- 3$ by $0$ .

$y = 0 + 12 (0) - 5$

Multiply $12$ by $0$ .

$y = 0 + 0 - 5$

Simplify by adding and subtracting.

Add $0$ and $0$ .

$y = 0 - 5$

Subtract $5$ from $0$ .

$y = - 5$

y-intercept(s) in point form.

y-intercept(s): $(0, - 5)$

Step 3

List the intersections.

x-intercept(s): $(\frac{6 + \sqrt{21}}{3}, 0), (\frac{6 - \sqrt{21}}{3}, 0)$

y-intercept(s): $(0, - 5)$

Step 4

Find the x and y Intercepts y=-3x^2+12x-5

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