Algebra Examples

$y = x^{2} - 2 \sqrt{2} x + 2$

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 $x$ .

$0 = x^{2} - 2 \sqrt{2} x + 2$

Step 1.2

Solve the equation.

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Step 1.2.1

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

$x^{2} - 2 \sqrt{2} x + 2 = 0$

Step 1.2.2

Use the quadratic formula to find the solutions.

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

Step 1.2.3

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

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

Step 1.2.4

Simplify.

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Step 1.2.4.1

Simplify the numerator.

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Step 1.2.4.1.1

Apply the product rule to $- 2 \sqrt{2}$ .

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

Step 1.2.4.1.2

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

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

Step 1.2.4.1.3

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.2.4.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Step 1.2.4.1.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.2.4.1.3.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 1.2.4.1.3.4

Cancel the common factor of $2$ .

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Step 1.2.4.1.3.4.1

Cancel the common factor.

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

Step 1.2.4.1.3.4.2

Rewrite the expression.

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

Step 1.2.4.1.3.5

Evaluate the exponent.

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

Step 1.2.4.1.4

Multiply $4$ by $2$ .

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

Step 1.2.4.1.5

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

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Step 1.2.4.1.5.1

Multiply $- 4$ by $1$ .

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

Step 1.2.4.1.5.2

Multiply $- 4$ by $2$ .

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

Step 1.2.4.1.6

Subtract $8$ from $8$ .

$x = \frac{2 \sqrt{2} \pm \sqrt{0}}{2 \cdot 1}$

Step 1.2.4.1.7

Rewrite $0$ as $0^{2}$ .

$x = \frac{2 \sqrt{2} \pm \sqrt{0^{2}}}{2 \cdot 1}$

Step 1.2.4.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{2 \sqrt{2} \pm 0}{2 \cdot 1}$

Step 1.2.4.1.9

$2 \sqrt{2}$ plus or minus $0$ is $2 \sqrt{2}$ .

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

Step 1.2.4.2

Multiply $2$ by $1$ .

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

Step 1.2.4.3

Cancel the common factor of $2$ .

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Step 1.2.4.3.1

Cancel the common factor.

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

Step 1.2.4.3.2

Divide $\sqrt{2}$ by $1$ .

$x = \sqrt{2}$

Step 1.2.5

The final answer is the combination of both solutions.

$x = \sqrt{2}$ Double roots

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

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

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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 $0$ .

$y = 0 - 2 \sqrt{2} \cdot 0 + 2$

Step 2.2.3.1.2

Multiply $- 2 \sqrt{2} \cdot 0$ .

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Step 2.2.3.1.2.1

Multiply $0$ by $- 2$ .

$y = 0 + 0 \sqrt{2} + 2$

Step 2.2.3.1.2.2

Multiply $0$ by $\sqrt{2}$ .

$y = 0 + 0 + 2$

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): $(\sqrt{2}, 0)$

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

Step 4