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

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

Step 1

Find the x-intercepts.

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

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

Solve the equation.

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

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

Factor $3$ out of $3 x^{2} + 3 x - 3$ .

Factor $3$ out of $3 x^{2}$ .

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

Factor $3$ out of $3 x$ .

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

Factor $3$ out of $- 3$ .

$3 (x^{2}) + 3 (x) + 3 (- 1) = 0$

Factor $3$ out of $3 (x^{2}) + 3 (x)$ .

$3 (x^{2} + x) + 3 (- 1) = 0$

Factor $3$ out of $3 (x^{2} + x) + 3 (- 1)$ .

$3 (x^{2} + x - 1) = 0$

Divide each term in $3 (x^{2} + x - 1) = 0$ by $3$ and simplify.

Divide each term in $3 (x^{2} + x - 1) = 0$ by $3$ .

$\frac{3 (x^{2} + x - 1)}{3} = \frac{0}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 (x^{2} + x - 1)}{3} = \frac{0}{3}$

Divide $x^{2} + x - 1$ by $1$ .

$x^{2} + x - 1 = \frac{0}{3}$

Simplify the right side.

Divide $0$ by $3$ .

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

Use the quadratic formula to find the solutions.

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

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

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

Simplify.

Simplify the numerator.

One to any power is one.

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

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

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 1$ .

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

Add $1$ and $4$ .

$x = \frac{- 1 \pm \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{5}}{2}$

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

Simplify the numerator.

One to any power is one.

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

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

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 1$ .

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

Add $1$ and $4$ .

$x = \frac{- 1 \pm \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{5}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{- 1 + \sqrt{5}}{2}$

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 + \sqrt{5}}{2}$

Factor $- 1$ out of $\sqrt{5}$ .

$x = \frac{- 1 \cdot 1 - 1 (- \sqrt{5})}{2}$

Factor $- 1$ out of $- 1 (1) - 1 (- \sqrt{5})$ .

$x = \frac{- 1 (1 - \sqrt{5})}{2}$

Move the negative in front of the fraction.

$x = - \frac{1 - \sqrt{5}}{2}$

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

Simplify the numerator.

One to any power is one.

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

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

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 1$ .

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

Add $1$ and $4$ .

$x = \frac{- 1 \pm \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{5}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{- 1 - \sqrt{5}}{2}$

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 - \sqrt{5}}{2}$

Factor $- 1$ out of $- \sqrt{5}$ .

$x = \frac{- 1 \cdot 1 - (\sqrt{5})}{2}$

Factor $- 1$ out of $- 1 (1) - (\sqrt{5})$ .

$x = \frac{- 1 (1 + \sqrt{5})}{2}$

Move the negative in front of the fraction.

$x = - \frac{1 + \sqrt{5}}{2}$

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{5}}{2}, - \frac{1 + \sqrt{5}}{2}$

x-intercept(s) in point form.

x-intercept(s): $(- \frac{1 - \sqrt{5}}{2}, 0), (- \frac{1 + \sqrt{5}}{2}, 0)$

Step 2

Find the y-intercepts.

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

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

Solve the equation.

Remove parentheses.

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

Remove parentheses.

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

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

Simplify each term.

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

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

Multiply $3$ by $0$ .

$y = 0 + 3 (0) - 3$

Multiply $3$ by $0$ .

$y = 0 + 0 - 3$

Simplify by adding and subtracting.

Add $0$ and $0$ .

$y = 0 - 3$

Subtract $3$ from $0$ .

$y = - 3$

y-intercept(s) in point form.

y-intercept(s): $(0, - 3)$

Step 3

List the intersections.

x-intercept(s): $(- \frac{1 - \sqrt{5}}{2}, 0), (- \frac{1 + \sqrt{5}}{2}, 0)$

y-intercept(s): $(0, - 3)$

Step 4

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

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