Pre-Algebra Examples

$\frac{1}{4} x^{2} + \frac{41}{4} = 3 x$

Step 1

Combine $\frac{1}{4}$ and $x^{2}$ .

$\frac{x^{2}}{4} + \frac{41}{4} = 3 x$

Step 2

Subtract $3 x$ from both sides of the equation.

$\frac{x^{2}}{4} + \frac{41}{4} - 3 x = 0$

Step 3

Multiply through by the least common denominator $4$ , then simplify.

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Step 3.1

Apply the distributive property.

$4 (\frac{x^{2}}{4}) + 4 (\frac{41}{4}) + 4 (- 3 x) = 0$

Step 3.2

Simplify.

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Step 3.2.1

Cancel the common factor of $4$ .

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Step 3.2.1.1

Cancel the common factor.

$4 (\frac{x^{2}}{4}) + 4 (\frac{41}{4}) + 4 (- 3 x) = 0$

Step 3.2.1.2

Rewrite the expression.

$x^{2} + 4 (\frac{41}{4}) + 4 (- 3 x) = 0$

Step 3.2.2

Cancel the common factor of $4$ .

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Step 3.2.2.1

Cancel the common factor.

$x^{2} + 4 (\frac{41}{4}) + 4 (- 3 x) = 0$

Step 3.2.2.2

Rewrite the expression.

$x^{2} + 41 + 4 (- 3 x) = 0$

Step 3.2.3

Multiply $- 3$ by $4$ .

$x^{2} + 41 - 12 x = 0$

Step 3.3

Move $41$ .

$x^{2} - 12 x + 41 = 0$

Step 4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5

Substitute the values $a = 1$ , $b = - 12$ , and $c = 41$ into the quadratic formula and solve for $x$ .

$\frac{12 \pm \sqrt{{(- 12)}^{2} - 4 \cdot (1 \cdot 41)}}{2 \cdot 1}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1 \cdot 41}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot 41$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 41}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $41$ .

$x = \frac{12 \pm \sqrt{144 - 164}}{2 \cdot 1}$

Step 6.1.3

Subtract $164$ from $144$ .

$x = \frac{12 \pm \sqrt{- 20}}{2 \cdot 1}$

Step 6.1.4

Rewrite $- 20$ as $- 1 (20)$ .

$x = \frac{12 \pm \sqrt{- 1 \cdot 20}}{2 \cdot 1}$

Step 6.1.5

Rewrite $\sqrt{- 1 (20)}$ as $\sqrt{- 1} \cdot \sqrt{20}$ .

$x = \frac{12 \pm \sqrt{- 1} \cdot \sqrt{20}}{2 \cdot 1}$

Step 6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{12 \pm i \cdot \sqrt{20}}{2 \cdot 1}$

Step 6.1.7

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 6.1.7.1

Factor $4$ out of $20$ .

$x = \frac{12 \pm i \cdot \sqrt{4 (5)}}{2 \cdot 1}$

Step 6.1.7.2

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

$x = \frac{12 \pm i \cdot \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Step 6.1.8

Pull terms out from under the radical.

$x = \frac{12 \pm i \cdot (2 \sqrt{5})}{2 \cdot 1}$

Step 6.1.9

Move $2$ to the left of $i$ .

$x = \frac{12 \pm 2 i \sqrt{5}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$x = \frac{12 \pm 2 i \sqrt{5}}{2}$

Step 6.3

Simplify $\frac{12 \pm 2 i \sqrt{5}}{2}$ .

$x = 6 \pm i \sqrt{5}$

Step 7

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1 \cdot 41}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot 41$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 41}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $41$ .

$x = \frac{12 \pm \sqrt{144 - 164}}{2 \cdot 1}$

Step 7.1.3

Subtract $164$ from $144$ .

$x = \frac{12 \pm \sqrt{- 20}}{2 \cdot 1}$

Step 7.1.4

Rewrite $- 20$ as $- 1 (20)$ .

$x = \frac{12 \pm \sqrt{- 1 \cdot 20}}{2 \cdot 1}$

Step 7.1.5

Rewrite $\sqrt{- 1 (20)}$ as $\sqrt{- 1} \cdot \sqrt{20}$ .

$x = \frac{12 \pm \sqrt{- 1} \cdot \sqrt{20}}{2 \cdot 1}$

Step 7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{12 \pm i \cdot \sqrt{20}}{2 \cdot 1}$

Step 7.1.7

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 7.1.7.1

Factor $4$ out of $20$ .

$x = \frac{12 \pm i \cdot \sqrt{4 (5)}}{2 \cdot 1}$

Step 7.1.7.2

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

$x = \frac{12 \pm i \cdot \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Step 7.1.8

Pull terms out from under the radical.

$x = \frac{12 \pm i \cdot (2 \sqrt{5})}{2 \cdot 1}$

Step 7.1.9

Move $2$ to the left of $i$ .

$x = \frac{12 \pm 2 i \sqrt{5}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$x = \frac{12 \pm 2 i \sqrt{5}}{2}$

Step 7.3

Simplify $\frac{12 \pm 2 i \sqrt{5}}{2}$ .

$x = 6 \pm i \sqrt{5}$

Step 7.4

Change the $\pm$ to $+$ .

$x = 6 + i \sqrt{5}$

Step 8

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1 \cdot 41}}{2 \cdot 1}$

Step 8.1.2

Multiply $- 4 \cdot 1 \cdot 41$ .

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Step 8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 41}}{2 \cdot 1}$

Step 8.1.2.2

Multiply $- 4$ by $41$ .

$x = \frac{12 \pm \sqrt{144 - 164}}{2 \cdot 1}$

Step 8.1.3

Subtract $164$ from $144$ .

$x = \frac{12 \pm \sqrt{- 20}}{2 \cdot 1}$

Step 8.1.4

Rewrite $- 20$ as $- 1 (20)$ .

$x = \frac{12 \pm \sqrt{- 1 \cdot 20}}{2 \cdot 1}$

Step 8.1.5

Rewrite $\sqrt{- 1 (20)}$ as $\sqrt{- 1} \cdot \sqrt{20}$ .

$x = \frac{12 \pm \sqrt{- 1} \cdot \sqrt{20}}{2 \cdot 1}$

Step 8.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{12 \pm i \cdot \sqrt{20}}{2 \cdot 1}$

Step 8.1.7

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 8.1.7.1

Factor $4$ out of $20$ .

$x = \frac{12 \pm i \cdot \sqrt{4 (5)}}{2 \cdot 1}$

Step 8.1.7.2

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

$x = \frac{12 \pm i \cdot \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Step 8.1.8

Pull terms out from under the radical.

$x = \frac{12 \pm i \cdot (2 \sqrt{5})}{2 \cdot 1}$

Step 8.1.9

Move $2$ to the left of $i$ .

$x = \frac{12 \pm 2 i \sqrt{5}}{2 \cdot 1}$

Step 8.2

Multiply $2$ by $1$ .

$x = \frac{12 \pm 2 i \sqrt{5}}{2}$

Step 8.3

Simplify $\frac{12 \pm 2 i \sqrt{5}}{2}$ .

$x = 6 \pm i \sqrt{5}$

Step 8.4

Change the $\pm$ to $-$ .

$x = 6 - i \sqrt{5}$

Step 9

The final answer is the combination of both solutions.

$x = 6 + i \sqrt{5}, 6 - i \sqrt{5}$