Algebra Examples

$x^{2} + {(y - 3 \sqrt{2 x})}^{2} = 1$

Step 1

Solve for $y$ .

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Step 1.1

Subtract $x^{2}$ from both sides of the equation.

${(y - 3 \sqrt{2 x})}^{2} = 1 - x^{2}$

Step 1.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y - 3 \sqrt{2 x} = \pm \sqrt{1 - x^{2}}$

Step 1.3

Simplify $\pm \sqrt{1 - x^{2}}$ .

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Step 1.3.1

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

$y - 3 \sqrt{2 x} = \pm \sqrt{1^{2} - x^{2}}$

Step 1.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

$y - 3 \sqrt{2 x} = \pm \sqrt{(1 + x) (1 - x)}$

Step 1.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.4.1

First, use the positive value of the $\pm$ to find the first solution.

$y - 3 \sqrt{2 x} = \sqrt{(1 + x) (1 - x)}$

Step 1.4.2

Add $3 \sqrt{2 x}$ to both sides of the equation.

$y = \sqrt{(1 + x) (1 - x)} + 3 \sqrt{2 x}$

Step 1.4.3

Next, use the negative value of the $\pm$ to find the second solution.

$y - 3 \sqrt{2 x} = - \sqrt{(1 + x) (1 - x)}$

Step 1.4.4

Add $3 \sqrt{2 x}$ to both sides of the equation.

$y = - \sqrt{(1 + x) (1 - x)} + 3 \sqrt{2 x}$

Step 1.4.5

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt{(1 + x) (1 - x)} + 3 \sqrt{2 x}$

$y = - \sqrt{(1 + x) (1 - x)} + 3 \sqrt{2 x}$

$y = \sqrt{(1 + x) (1 - x)} + 3 \sqrt{2 x}$

$y = - \sqrt{(1 + x) (1 - x)} + 3 \sqrt{2 x}$

$y = \sqrt{(1 + x) (1 - x)} + 3 \sqrt{2 x}$

$y = - \sqrt{(1 + x) (1 - x)} + 3 \sqrt{2 x}$

Step 2

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$y = a x^{2} + b x + c$

Step 3

The standard form is $\sqrt{(1 + x) (1 - x)} + 3 \sqrt{2 x}, - \sqrt{(1 + x) (1 - x)} + 3 \sqrt{2 x}$ .

$y = \sqrt{(1 + x) (1 - x)} + 3 \sqrt{2 x}$

$y = - \sqrt{(1 + x) (1 - x)} + 3 \sqrt{2 x}$

Step 4