Linear Algebra Examples

$2 m^{2} + 3 = m$

Step 1

Subtract $m$ from both sides of the equation.

$2 m^{2} + 3 - m = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 2$ , $b = - 1$ , and $c = 3$ into the quadratic formula and solve for $m$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (2 \cdot 3)}}{2 \cdot 2}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$m = \frac{1 \pm \sqrt{1 - 4 \cdot 2 \cdot 3}}{2 \cdot 2}$

Step 4.1.2

Multiply $- 4 \cdot 2 \cdot 3$ .

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Step 4.1.2.1

Multiply $- 4$ by $2$ .

$m = \frac{1 \pm \sqrt{1 - 8 \cdot 3}}{2 \cdot 2}$

Step 4.1.2.2

Multiply $- 8$ by $3$ .

$m = \frac{1 \pm \sqrt{1 - 24}}{2 \cdot 2}$

Step 4.1.3

Subtract $24$ from $1$ .

$m = \frac{1 \pm \sqrt{- 23}}{2 \cdot 2}$

Step 4.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$m = \frac{1 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 2}$

Step 4.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$m = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 2}$

Step 4.1.6

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

$m = \frac{1 \pm i \sqrt{23}}{2 \cdot 2}$

Step 4.2

Multiply $2$ by $2$ .

$m = \frac{1 \pm i \sqrt{23}}{4}$

Step 5

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

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$m = \frac{1 \pm \sqrt{1 - 4 \cdot 2 \cdot 3}}{2 \cdot 2}$

Step 5.1.2

Multiply $- 4 \cdot 2 \cdot 3$ .

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Step 5.1.2.1

Multiply $- 4$ by $2$ .

$m = \frac{1 \pm \sqrt{1 - 8 \cdot 3}}{2 \cdot 2}$

Step 5.1.2.2

Multiply $- 8$ by $3$ .

$m = \frac{1 \pm \sqrt{1 - 24}}{2 \cdot 2}$

Step 5.1.3

Subtract $24$ from $1$ .

$m = \frac{1 \pm \sqrt{- 23}}{2 \cdot 2}$

Step 5.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$m = \frac{1 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 2}$

Step 5.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$m = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 2}$

Step 5.1.6

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

$m = \frac{1 \pm i \sqrt{23}}{2 \cdot 2}$

Step 5.2

Multiply $2$ by $2$ .

$m = \frac{1 \pm i \sqrt{23}}{4}$

Step 5.3

Change the $\pm$ to $+$ .

$m = \frac{1 + i \sqrt{23}}{4}$

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

$m = \frac{1 \pm \sqrt{1 - 4 \cdot 2 \cdot 3}}{2 \cdot 2}$

Step 6.1.2

Multiply $- 4 \cdot 2 \cdot 3$ .

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Step 6.1.2.1

Multiply $- 4$ by $2$ .

$m = \frac{1 \pm \sqrt{1 - 8 \cdot 3}}{2 \cdot 2}$

Step 6.1.2.2

Multiply $- 8$ by $3$ .

$m = \frac{1 \pm \sqrt{1 - 24}}{2 \cdot 2}$

Step 6.1.3

Subtract $24$ from $1$ .

$m = \frac{1 \pm \sqrt{- 23}}{2 \cdot 2}$

Step 6.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$m = \frac{1 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 2}$

Step 6.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$m = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 2}$

Step 6.1.6

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

$m = \frac{1 \pm i \sqrt{23}}{2 \cdot 2}$

Step 6.2

Multiply $2$ by $2$ .

$m = \frac{1 \pm i \sqrt{23}}{4}$

Step 6.3

Change the $\pm$ to $-$ .

$m = \frac{1 - i \sqrt{23}}{4}$

Step 7

The final answer is the combination of both solutions.

$m = \frac{1 + i \sqrt{23}}{4}, \frac{1 - i \sqrt{23}}{4}$

Step 8

The domain is the set of all valid $m$ values.

${m | m = \frac{1 + i \sqrt{23}}{4}, \frac{1 - i \sqrt{23}}{4}}$

Step 9