Finite Math Examples

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

Step 1

Subtract $2 x$ from both sides of the equation.

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

Step 2

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

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

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 $\sqrt{{(x - 1)}^{2}}$ as ${(x - 1)}^{\frac{2}{2}}$ .

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

Step 3.2

Divide $2$ by $2$ .

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

Step 3.3

Simplify the left side.

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Step 3.3.1

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

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Step 3.3.1.1

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

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

Step 3.3.1.2

Multiply $2$ by $1$ .

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

Step 3.4

Simplify the right side.

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Step 3.4.1

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

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Step 3.4.1.1

Apply the product rule to $- 2 x$ .

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

Step 3.4.1.2

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

${(x - 1)}^{2} = 4 x^{2}$

Step 4

Solve for $x$ .

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Step 4.1

Move all terms containing $x$ to the left side of the equation.

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Step 4.1.1

Subtract $4 x^{2}$ from both sides of the equation.

${(x - 1)}^{2} - 4 x^{2} = 0$

Step 4.1.2

Simplify each term.

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Step 4.1.2.1

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$(x - 1) (x - 1) - 4 x^{2} = 0$

Step 4.1.2.2

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 4.1.2.2.1

Apply the distributive property.

$x (x - 1) - 1 (x - 1) - 4 x^{2} = 0$

Step 4.1.2.2.2

Apply the distributive property.

$x \cdot x + x \cdot - 1 - 1 (x - 1) - 4 x^{2} = 0$

Step 4.1.2.2.3

Apply the distributive property.

$x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1 - 4 x^{2} = 0$

Step 4.1.2.3

Simplify and combine like terms.

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Step 4.1.2.3.1

Simplify each term.

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Step 4.1.2.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1 - 4 x^{2} = 0$

Step 4.1.2.3.1.2

Move $- 1$ to the left of $x$ .

$x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1 - 4 x^{2} = 0$

Step 4.1.2.3.1.3

Rewrite $- 1 x$ as $- x$ .

$x^{2} - x - 1 x - 1 \cdot - 1 - 4 x^{2} = 0$

Step 4.1.2.3.1.4

Rewrite $- 1 x$ as $- x$ .

$x^{2} - x - x - 1 \cdot - 1 - 4 x^{2} = 0$

Step 4.1.2.3.1.5

Multiply $- 1$ by $- 1$ .

$x^{2} - x - x + 1 - 4 x^{2} = 0$

Step 4.1.2.3.2

Subtract $x$ from $- x$ .

$x^{2} - 2 x + 1 - 4 x^{2} = 0$

Step 4.1.3

Subtract $4 x^{2}$ from $x^{2}$ .

$- 3 x^{2} - 2 x + 1 = 0$

Step 4.2

Factor the left side of the equation.

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Step 4.2.1

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

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Step 4.2.1.1

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

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

Step 4.2.1.2

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

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

Step 4.2.1.3

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

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

Step 4.2.1.4

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

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

Step 4.2.1.5

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

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

Step 4.2.2

Factor.

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Step 4.2.2.1

Factor by grouping.

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Step 4.2.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 1 = - 3$ and whose sum is $b = 2$ .

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Step 4.2.2.1.1.1

Factor $2$ out of $2 x$ .

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

Step 4.2.2.1.1.2

Rewrite $2$ as $- 1$ plus $3$

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

Step 4.2.2.1.1.3

Apply the distributive property.

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

Step 4.2.2.1.2

Factor out the greatest common factor from each group.

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Step 4.2.2.1.2.1

Group the first two terms and the last two terms.

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

Step 4.2.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 4.2.2.1.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

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

Step 4.2.2.2

Remove unnecessary parentheses.

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

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 $0$ .

$3 x - 1 = 0$

$x + 1 = 0$

Step 4.4

Set $3 x - 1$ equal to $0$ and solve for $x$ .

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Step 4.4.1

Set $3 x - 1$ equal to $0$ .

$3 x - 1 = 0$

Step 4.4.2

Solve $3 x - 1 = 0$ for $x$ .

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Step 4.4.2.1

Add $1$ to both sides of the equation.

$3 x = 1$

Step 4.4.2.2

Divide each term in $3 x = 1$ by $3$ and simplify.

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Step 4.4.2.2.1

Divide each term in $3 x = 1$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3}$

Step 4.4.2.2.2

Simplify the left side.

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Step 4.4.2.2.2.1

Cancel the common factor of $3$ .

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Step 4.4.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{1}{3}$

Step 4.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{3}$

Step 4.5

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

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Step 4.5.1

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

$x + 1 = 0$

Step 4.5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 4.6

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

$x = \frac{1}{3}, - 1$

Step 5

Exclude the solutions that do not make $\sqrt{{(x - 1)}^{2}} + 2 x = 0$ true.

$x = - 1$

Step 6