Algebra Examples

$\sqrt{{(2 x - 3)}^{2}} = 7$

Step 1

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

${\sqrt{{(2 x - 3)}^{2}}}^{2} = 7^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

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

${({(2 x - 3)}^{\frac{2}{2}})}^{2} = 7^{2}$

Step 2.2

Divide $2$ by $2$ .

${({(2 x - 3)}^{1})}^{2} = 7^{2}$

Step 2.3

Simplify the left side.

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Step 2.3.1

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

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Step 2.3.1.1

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

${(2 x - 3)}^{1 \cdot 2} = 7^{2}$

Step 2.3.1.2

Multiply $2$ by $1$ .

${(2 x - 3)}^{2} = 7^{2}$

Step 2.4

Simplify the right side.

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Step 2.4.1

Raise $7$ to the power of $2$ .

${(2 x - 3)}^{2} = 49$

Step 3

Solve for $x$ .

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Step 3.1

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

$2 x - 3 = \pm \sqrt{49}$

Step 3.2

Simplify $\pm \sqrt{49}$ .

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Step 3.2.1

Rewrite $49$ as $7^{2}$ .

$2 x - 3 = \pm \sqrt{7^{2}}$

Step 3.2.2

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

$2 x - 3 = \pm 7$

Step 3.3

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

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Step 3.3.1

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

$2 x - 3 = 7$

Step 3.3.2

Move all terms not containing $x$ to the right side of the equation.

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Step 3.3.2.1

Add $3$ to both sides of the equation.

$2 x = 7 + 3$

Step 3.3.2.2

Add $7$ and $3$ .

$2 x = 10$

Step 3.3.3

Divide each term in $2 x = 10$ by $2$ and simplify.

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Step 3.3.3.1

Divide each term in $2 x = 10$ by $2$ .

$\frac{2 x}{2} = \frac{10}{2}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{10}{2}$

Step 3.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{10}{2}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Divide $10$ by $2$ .

$x = 5$

Step 3.3.4

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

$2 x - 3 = - 7$

Step 3.3.5

Move all terms not containing $x$ to the right side of the equation.

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Step 3.3.5.1

Add $3$ to both sides of the equation.

$2 x = - 7 + 3$

Step 3.3.5.2

Add $- 7$ and $3$ .

$2 x = - 4$

Step 3.3.6

Divide each term in $2 x = - 4$ by $2$ and simplify.

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Step 3.3.6.1

Divide each term in $2 x = - 4$ by $2$ .

$\frac{2 x}{2} = \frac{- 4}{2}$

Step 3.3.6.2

Simplify the left side.

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Step 3.3.6.2.1

Cancel the common factor of $2$ .

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Step 3.3.6.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 4}{2}$

Step 3.3.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{2}$

Step 3.3.6.3

Simplify the right side.

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Step 3.3.6.3.1

Divide $- 4$ by $2$ .

$x = - 2$

Step 3.3.7

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

$x = 5, - 2$