Find the x and y Intercepts 3x^2-12x+6

$3 x^{2} - 12 x + 6$

Step 1

Write $3 x^{2} - 12 x + 6$ as an equation.

$y = 3 x^{2} - 12 x + 6$

Step 2

Find the x-intercepts.

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

$0 = 3 x^{2} - 12 x + 6$

Solve the equation.

Rewrite the equation as $3 x^{2} - 12 x + 6 = 0$ .

$3 x^{2} - 12 x + 6 = 0$

Factor $3$ out of $3 x^{2} - 12 x + 6$ .

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

$3 (x^{2}) - 12 x + 6 = 0$

Factor $3$ out of $- 12 x$ .

$3 (x^{2}) + 3 (- 4 x) + 6 = 0$

Factor $3$ out of $6$ .

$3 x^{2} + 3 (- 4 x) + 3 \cdot 2 = 0$

Factor $3$ out of $3 x^{2} + 3 (- 4 x)$ .

$3 (x^{2} - 4 x) + 3 \cdot 2 = 0$

Factor $3$ out of $3 (x^{2} - 4 x) + 3 \cdot 2$ .

$3 (x^{2} - 4 x + 2) = 0$

Divide each term in $3 (x^{2} - 4 x + 2) = 0$ by $3$ and simplify.

Divide each term in $3 (x^{2} - 4 x + 2) = 0$ by $3$ .

$\frac{3 (x^{2} - 4 x + 2)}{3} = \frac{0}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 (x^{2} - 4 x + 2)}{3} = \frac{0}{3}$

Divide $x^{2} - 4 x + 2$ by $1$ .

$x^{2} - 4 x + 2 = \frac{0}{3}$

Simplify the right side.

Divide $0$ by $3$ .

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

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 4$ , and $c = 2$ into the quadratic formula and solve for $x$ .

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

Simplify.

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

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

Multiply $- 4 \cdot 1 \cdot 2$ .

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $2$ .

$x = \frac{4 \pm \sqrt{16 - 8}}{2 \cdot 1}$

Subtract $8$ from $16$ .

$x = \frac{4 \pm \sqrt{8}}{2 \cdot 1}$

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 \sqrt{2}}{2}$

Simplify $\frac{4 \pm 2 \sqrt{2}}{2}$ .

$x = 2 \pm \sqrt{2}$

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

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

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

Multiply $- 4 \cdot 1 \cdot 2$ .

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $2$ .

$x = \frac{4 \pm \sqrt{16 - 8}}{2 \cdot 1}$

Subtract $8$ from $16$ .

$x = \frac{4 \pm \sqrt{8}}{2 \cdot 1}$

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 \sqrt{2}}{2}$

Simplify $\frac{4 \pm 2 \sqrt{2}}{2}$ .

$x = 2 \pm \sqrt{2}$

Change the $\pm$ to $+$ .

$x = 2 + \sqrt{2}$

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

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

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

Multiply $- 4 \cdot 1 \cdot 2$ .

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $2$ .

$x = \frac{4 \pm \sqrt{16 - 8}}{2 \cdot 1}$

Subtract $8$ from $16$ .

$x = \frac{4 \pm \sqrt{8}}{2 \cdot 1}$

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 \sqrt{2}}{2}$

Simplify $\frac{4 \pm 2 \sqrt{2}}{2}$ .

$x = 2 \pm \sqrt{2}$

Change the $\pm$ to $-$ .

$x = 2 - \sqrt{2}$

The final answer is the combination of both solutions.

$x = 2 + \sqrt{2}, 2 - \sqrt{2}$

x-intercept(s) in point form.

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

Step 3

Find the y-intercepts.

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

$y = 3 {(0)}^{2} - 12 \cdot 0 + 6$

Solve the equation.

Remove parentheses.

$y = 3 \cdot 0^{2} - 12 \cdot 0 + 6$

Remove parentheses.

$y = 3 {(0)}^{2} - 12 \cdot 0 + 6$

Simplify $3 {(0)}^{2} - 12 \cdot 0 + 6$ .

Simplify each term.

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

$y = 3 \cdot 0 - 12 \cdot 0 + 6$

Multiply $3$ by $0$ .

$y = 0 - 12 \cdot 0 + 6$

Multiply $- 12$ by $0$ .

$y = 0 + 0 + 6$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 6$

Add $0$ and $6$ .

$y = 6$

y-intercept(s) in point form.

y-intercept(s): $(0, 6)$

Step 4

List the intersections.

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

y-intercept(s): $(0, 6)$

Step 5

Find the x and y Intercepts 3x^2-12x+6

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