Algebra Examples

$9 {(7 - 5 x)}^{2} - 3 = 22$

Step 1

Add $3$ to both sides of the equation.

$9 {(7 - 5 x)}^{2} = 22 + 3$

Step 2

Add $22$ and $3$ .

$9 {(7 - 5 x)}^{2} = 25$

Step 3

Divide each term in $9 {(7 - 5 x)}^{2} = 25$ by $9$ and simplify.

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Step 3.1

Divide each term in $9 {(7 - 5 x)}^{2} = 25$ by $9$ .

$\frac{9 {(7 - 5 x)}^{2}}{9} = \frac{25}{9}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $9$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{9 {(7 - 5 x)}^{2}}{9} = \frac{25}{9}$

Step 3.2.1.2

Divide ${(7 - 5 x)}^{2}$ by $1$ .

${(7 - 5 x)}^{2} = \frac{25}{9}$

Step 4

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

$7 - 5 x = \pm \sqrt{\frac{25}{9}}$

Step 5

Simplify $\pm \sqrt{\frac{25}{9}}$ .

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Step 5.1

Rewrite $\sqrt{\frac{25}{9}}$ as $\frac{\sqrt{25}}{\sqrt{9}}$ .

$7 - 5 x = \pm \frac{\sqrt{25}}{\sqrt{9}}$

Step 5.2

Simplify the numerator.

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Step 5.2.1

Rewrite $25$ as $5^{2}$ .

$7 - 5 x = \pm \frac{\sqrt{5^{2}}}{\sqrt{9}}$

Step 5.2.2

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

$7 - 5 x = \pm \frac{5}{\sqrt{9}}$

Step 5.3

Simplify the denominator.

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Step 5.3.1

Rewrite $9$ as $3^{2}$ .

$7 - 5 x = \pm \frac{5}{\sqrt{3^{2}}}$

Step 5.3.2

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

$7 - 5 x = \pm \frac{5}{3}$

Step 6

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

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Step 6.1

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

$7 - 5 x = \frac{5}{3}$

Step 6.2

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

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Step 6.2.1

Subtract $7$ from both sides of the equation.

$- 5 x = \frac{5}{3} - 7$

Step 6.2.2

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 5 x = \frac{5}{3} - 7 \cdot \frac{3}{3}$

Step 6.2.3

Combine $- 7$ and $\frac{3}{3}$ .

$- 5 x = \frac{5}{3} + \frac{- 7 \cdot 3}{3}$

Step 6.2.4

Combine the numerators over the common denominator.

$- 5 x = \frac{5 - 7 \cdot 3}{3}$

Step 6.2.5

Simplify the numerator.

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Step 6.2.5.1

Multiply $- 7$ by $3$ .

$- 5 x = \frac{5 - 21}{3}$

Step 6.2.5.2

Subtract $21$ from $5$ .

$- 5 x = \frac{- 16}{3}$

Step 6.2.6

Move the negative in front of the fraction.

$- 5 x = - \frac{16}{3}$

Step 6.3

Divide each term in $- 5 x = - \frac{16}{3}$ by $- 5$ and simplify.

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Step 6.3.1

Divide each term in $- 5 x = - \frac{16}{3}$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{- \frac{16}{3}}{- 5}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $- 5$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{- \frac{16}{3}}{- 5}$

Step 6.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{16}{3}}{- 5}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{16}{3} \cdot \frac{1}{- 5}$

Step 6.3.3.2

Move the negative in front of the fraction.

$x = - \frac{16}{3} (- \frac{1}{5})$

Step 6.3.3.3

Multiply $- \frac{16}{3} (- \frac{1}{5})$ .

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Step 6.3.3.3.1

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{16}{3}) \frac{1}{5}$

Step 6.3.3.3.2

Multiply $\frac{16}{3}$ by $1$ .

$x = \frac{16}{3} \cdot \frac{1}{5}$

Step 6.3.3.3.3

Multiply $\frac{16}{3}$ by $\frac{1}{5}$ .

$x = \frac{16}{3 \cdot 5}$

Step 6.3.3.3.4

Multiply $3$ by $5$ .

$x = \frac{16}{15}$

Step 6.4

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

$7 - 5 x = - \frac{5}{3}$

Step 6.5

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

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Step 6.5.1

Subtract $7$ from both sides of the equation.

$- 5 x = - \frac{5}{3} - 7$

Step 6.5.2

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 5 x = - \frac{5}{3} - 7 \cdot \frac{3}{3}$

Step 6.5.3

Combine $- 7$ and $\frac{3}{3}$ .

$- 5 x = - \frac{5}{3} + \frac{- 7 \cdot 3}{3}$

Step 6.5.4

Combine the numerators over the common denominator.

$- 5 x = \frac{- 5 - 7 \cdot 3}{3}$

Step 6.5.5

Simplify the numerator.

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Step 6.5.5.1

Multiply $- 7$ by $3$ .

$- 5 x = \frac{- 5 - 21}{3}$

Step 6.5.5.2

Subtract $21$ from $- 5$ .

$- 5 x = \frac{- 26}{3}$

Step 6.5.6

Move the negative in front of the fraction.

$- 5 x = - \frac{26}{3}$

Step 6.6

Divide each term in $- 5 x = - \frac{26}{3}$ by $- 5$ and simplify.

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Step 6.6.1

Divide each term in $- 5 x = - \frac{26}{3}$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{- \frac{26}{3}}{- 5}$

Step 6.6.2

Simplify the left side.

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Step 6.6.2.1

Cancel the common factor of $- 5$ .

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Step 6.6.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{- \frac{26}{3}}{- 5}$

Step 6.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{26}{3}}{- 5}$

Step 6.6.3

Simplify the right side.

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Step 6.6.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{26}{3} \cdot \frac{1}{- 5}$

Step 6.6.3.2

Move the negative in front of the fraction.

$x = - \frac{26}{3} (- \frac{1}{5})$

Step 6.6.3.3

Multiply $- \frac{26}{3} (- \frac{1}{5})$ .

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Step 6.6.3.3.1

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{26}{3}) \frac{1}{5}$

Step 6.6.3.3.2

Multiply $\frac{26}{3}$ by $1$ .

$x = \frac{26}{3} \cdot \frac{1}{5}$

Step 6.6.3.3.3

Multiply $\frac{26}{3}$ by $\frac{1}{5}$ .

$x = \frac{26}{3 \cdot 5}$

Step 6.6.3.3.4

Multiply $3$ by $5$ .

$x = \frac{26}{15}$

Step 6.7

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

$x = \frac{16}{15}, \frac{26}{15}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{16}{15}, \frac{26}{15}$

Decimal Form:

$x = 1.0\overline{6}, 1.7\overline{3}$

Mixed Number Form:

$x = 1 \frac{1}{15}, 1 \frac{11}{15}$