Algebra Examples

$2 m^{2} - m - 15 = 0$

Step 1

Add $15$ to both sides of the equation.

$2 m^{2} - m = 15$

Step 2

Divide each term in $2 m^{2} - m = 15$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 m^{2} - m = 15$ by $2$ .

$\frac{2 m^{2}}{2} + \frac{- m}{2} = \frac{15}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Cancel the common factor of $2$ .

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Step 2.2.1.1.1

Cancel the common factor.

$\frac{2 m^{2}}{2} + \frac{- m}{2} = \frac{15}{2}$

Step 2.2.1.1.2

Divide $m^{2}$ by $1$ .

$m^{2} + \frac{- m}{2} = \frac{15}{2}$

Step 2.2.1.2

Move the negative in front of the fraction.

$m^{2} - \frac{m}{2} = \frac{15}{2}$

Step 3

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(- \frac{1}{4})}^{2}$

Step 4

Add the term to each side of the equation.

$m^{2} - \frac{m}{2} + {(- \frac{1}{4})}^{2} = \frac{15}{2} + {(- \frac{1}{4})}^{2}$

Step 5

Simplify the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify each term.

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Step 5.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.1.1.1.1

Apply the product rule to $- \frac{1}{4}$ .

$m^{2} - \frac{m}{2} + {(- 1)}^{2} {(\frac{1}{4})}^{2} = \frac{15}{2} + {(- \frac{1}{4})}^{2}$

Step 5.1.1.1.2

Apply the product rule to $\frac{1}{4}$ .

$m^{2} - \frac{m}{2} + {(- 1)}^{2} \frac{1^{2}}{4^{2}} = \frac{15}{2} + {(- \frac{1}{4})}^{2}$

Step 5.1.1.2

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

$m^{2} - \frac{m}{2} + 1 \frac{1^{2}}{4^{2}} = \frac{15}{2} + {(- \frac{1}{4})}^{2}$

Step 5.1.1.3

Multiply $\frac{1^{2}}{4^{2}}$ by $1$ .

$m^{2} - \frac{m}{2} + \frac{1^{2}}{4^{2}} = \frac{15}{2} + {(- \frac{1}{4})}^{2}$

Step 5.1.1.4

One to any power is one.

$m^{2} - \frac{m}{2} + \frac{1}{4^{2}} = \frac{15}{2} + {(- \frac{1}{4})}^{2}$

Step 5.1.1.5

Raise $4$ to the power of $2$ .

$m^{2} - \frac{m}{2} + \frac{1}{16} = \frac{15}{2} + {(- \frac{1}{4})}^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{15}{2} + {(- \frac{1}{4})}^{2}$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.2.1.1.1.1

Apply the product rule to $- \frac{1}{4}$ .

$m^{2} - \frac{m}{2} + \frac{1}{16} = \frac{15}{2} + {(- 1)}^{2} {(\frac{1}{4})}^{2}$

Step 5.2.1.1.1.2

Apply the product rule to $\frac{1}{4}$ .

$m^{2} - \frac{m}{2} + \frac{1}{16} = \frac{15}{2} + {(- 1)}^{2} \frac{1^{2}}{4^{2}}$

Step 5.2.1.1.2

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

$m^{2} - \frac{m}{2} + \frac{1}{16} = \frac{15}{2} + 1 \frac{1^{2}}{4^{2}}$

Step 5.2.1.1.3

Multiply $\frac{1^{2}}{4^{2}}$ by $1$ .

$m^{2} - \frac{m}{2} + \frac{1}{16} = \frac{15}{2} + \frac{1^{2}}{4^{2}}$

Step 5.2.1.1.4

One to any power is one.

$m^{2} - \frac{m}{2} + \frac{1}{16} = \frac{15}{2} + \frac{1}{4^{2}}$

Step 5.2.1.1.5

Raise $4$ to the power of $2$ .

$m^{2} - \frac{m}{2} + \frac{1}{16} = \frac{15}{2} + \frac{1}{16}$

Step 5.2.1.2

To write $\frac{15}{2}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$m^{2} - \frac{m}{2} + \frac{1}{16} = \frac{15}{2} \cdot \frac{8}{8} + \frac{1}{16}$

Step 5.2.1.3

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.1.3.1

Multiply $\frac{15}{2}$ by $\frac{8}{8}$ .

$m^{2} - \frac{m}{2} + \frac{1}{16} = \frac{15 \cdot 8}{2 \cdot 8} + \frac{1}{16}$

Step 5.2.1.3.2

Multiply $2$ by $8$ .

$m^{2} - \frac{m}{2} + \frac{1}{16} = \frac{15 \cdot 8}{16} + \frac{1}{16}$

Step 5.2.1.4

Combine the numerators over the common denominator.

$m^{2} - \frac{m}{2} + \frac{1}{16} = \frac{15 \cdot 8 + 1}{16}$

Step 5.2.1.5

Simplify the numerator.

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Step 5.2.1.5.1

Multiply $15$ by $8$ .

$m^{2} - \frac{m}{2} + \frac{1}{16} = \frac{120 + 1}{16}$

Step 5.2.1.5.2

Add $120$ and $1$ .

$m^{2} - \frac{m}{2} + \frac{1}{16} = \frac{121}{16}$

Step 6

Factor the perfect trinomial square into ${(m - \frac{1}{4})}^{2}$ .

${(m - \frac{1}{4})}^{2} = \frac{121}{16}$

Step 7

Solve the equation for $m$ .

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Step 7.1

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

$m - \frac{1}{4} = \pm \sqrt{\frac{121}{16}}$

Step 7.2

Simplify $\pm \sqrt{\frac{121}{16}}$ .

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Step 7.2.1

Rewrite $\sqrt{\frac{121}{16}}$ as $\frac{\sqrt{121}}{\sqrt{16}}$ .

$m - \frac{1}{4} = \pm \frac{\sqrt{121}}{\sqrt{16}}$

Step 7.2.2

Simplify the numerator.

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Step 7.2.2.1

Rewrite $121$ as $11^{2}$ .

$m - \frac{1}{4} = \pm \frac{\sqrt{11^{2}}}{\sqrt{16}}$

Step 7.2.2.2

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

$m - \frac{1}{4} = \pm \frac{11}{\sqrt{16}}$

Step 7.2.3

Simplify the denominator.

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Step 7.2.3.1

Rewrite $16$ as $4^{2}$ .

$m - \frac{1}{4} = \pm \frac{11}{\sqrt{4^{2}}}$

Step 7.2.3.2

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

$m - \frac{1}{4} = \pm \frac{11}{4}$

Step 7.3

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

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Step 7.3.1

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

$m - \frac{1}{4} = \frac{11}{4}$

Step 7.3.2

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

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Step 7.3.2.1

Add $\frac{1}{4}$ to both sides of the equation.

$m = \frac{11}{4} + \frac{1}{4}$

Step 7.3.2.2

Combine the numerators over the common denominator.

$m = \frac{11 + 1}{4}$

Step 7.3.2.3

Add $11$ and $1$ .

$m = \frac{12}{4}$

Step 7.3.2.4

Divide $12$ by $4$ .

$m = 3$

Step 7.3.3

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

$m - \frac{1}{4} = - \frac{11}{4}$

Step 7.3.4

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

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Step 7.3.4.1

Add $\frac{1}{4}$ to both sides of the equation.

$m = - \frac{11}{4} + \frac{1}{4}$

Step 7.3.4.2

Combine the numerators over the common denominator.

$m = \frac{- 11 + 1}{4}$

Step 7.3.4.3

Add $- 11$ and $1$ .

$m = \frac{- 10}{4}$

Step 7.3.4.4

Cancel the common factor of $- 10$ and $4$ .

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Step 7.3.4.4.1

Factor $2$ out of $- 10$ .

$m = \frac{2 (- 5)}{4}$

Step 7.3.4.4.2

Cancel the common factors.

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Step 7.3.4.4.2.1

Factor $2$ out of $4$ .

$m = \frac{2 \cdot - 5}{2 \cdot 2}$

Step 7.3.4.4.2.2

Cancel the common factor.

$m = \frac{2 \cdot - 5}{2 \cdot 2}$

Step 7.3.4.4.2.3

Rewrite the expression.

$m = \frac{- 5}{2}$

Step 7.3.4.5

Move the negative in front of the fraction.

$m = - \frac{5}{2}$

Step 7.3.5

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

$m = 3, - \frac{5}{2}$