Precalculus Examples

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

Step 1

Find the x-intercepts.

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Step 1.1

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

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

Step 1.2

Solve the equation.

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Step 1.2.1

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

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

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

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

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Step 1.2.2.1.1

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

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

Step 1.2.2.1.2

Factor $2$ out of $- 4 x$ .

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

Step 1.2.2.1.3

Factor $2$ out of $2$ .

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

Step 1.2.2.1.4

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

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

Step 1.2.2.1.5

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

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

Step 1.2.2.2

Factor using the perfect square rule.

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Step 1.2.2.2.1

Rewrite $1$ as $1^{2}$ .

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

Step 1.2.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 1.2.2.2.3

Rewrite the polynomial.

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

Step 1.2.2.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

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

Step 1.2.3

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

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Step 1.2.3.1

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

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

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $2$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

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

Step 1.2.3.2.1.2

Divide ${(x - 1)}^{2}$ by $1$ .

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

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $0$ by $2$ .

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

Step 1.2.4

Set the $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 1.2.5

Add $1$ to both sides of the equation.

$x = 1$

Step 1.3

x-intercept(s) in point form.

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

Step 2

Find the y-intercepts.

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Step 2.1

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

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

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

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

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

$y = 2 \cdot 0 - 4 \cdot 0 + 2$

Step 2.2.3.1.2

Multiply $2$ by $0$ .

$y = 0 - 4 \cdot 0 + 2$

Step 2.2.3.1.3

Multiply $- 4$ by $0$ .

$y = 0 + 0 + 2$

Step 2.2.3.2

Simplify by adding numbers.

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Step 2.2.3.2.1

Add $0$ and $0$ .

$y = 0 + 2$

Step 2.2.3.2.2

Add $0$ and $2$ .

$y = 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 2)$

Step 3

List the intersections.

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

y-intercept(s): $(0, 2)$

Step 4