Algebra Examples

$x + \sqrt{3 - 2 x} = 0$

Step 1

Subtract $x$ from both sides of the equation.

$\sqrt{3 - 2 x} = - x$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

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

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 - 2 x}$ as ${(3 - 2 x)}^{\frac{1}{2}}$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(3 - 2 x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(3 - 2 x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.1.2

Simplify.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(- x)}^{2}$ .

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Step 3.3.1.1

Apply the product rule to $- x$ .

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

Step 3.3.1.2

Raise $- 1$ to the power of $2$ .

$3 - 2 x = 1 x^{2}$

Step 3.3.1.3

Multiply $x^{2}$ by $1$ .

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

Step 4

Solve for $x$ .

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Step 4.1

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

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

Step 4.2

Factor the left side of the equation.

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Step 4.2.1

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

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Step 4.2.1.1

Reorder the expression.

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Step 4.2.1.1.1

Move $3$ .

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

Step 4.2.1.1.2

Reorder $- 2 x$ and $- x^{2}$ .

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

Step 4.2.1.2

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

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

Step 4.2.1.3

Factor $- 1$ out of $- 2 x$ .

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

Step 4.2.1.4

Rewrite $3$ as $- 1 (- 3)$ .

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

Step 4.2.1.5

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

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

Step 4.2.1.6

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

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

Step 4.2.2

Factor.

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Step 4.2.2.1

Factor $x^{2} + 2 x - 3$ using the AC method.

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Step 4.2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3$ and whose sum is $2$ .

$- 1, 3$

Step 4.2.2.1.2

Write the factored form using these integers.

$- ((x - 1) (x + 3)) = 0$

Step 4.2.2.2

Remove unnecessary parentheses.

$- (x - 1) (x + 3) = 0$

Step 4.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 1 = 0$

$x + 3 = 0$

Step 4.4

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 4.4.1

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

$x - 1 = 0$

Step 4.4.2

Add $1$ to both sides of the equation.

$x = 1$

Step 4.5

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 4.5.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 4.5.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 4.6

The final solution is all the values that make $- (x - 1) (x + 3) = 0$ true.

$x = 1, - 3$

Step 5

Exclude the solutions that do not make $x + \sqrt{3 - 2 x} = 0$ true.

$x = - 3$