Algebra Examples

$4 m^{2} - 3 m = n$

Step 1

Solve the equation for $m$ .

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Step 1.1

Subtract $n$ from both sides of the equation.

$4 m^{2} - 3 m - n = 0$

Step 1.2

Use the quadratic formula to find the solutions.

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

Step 1.3

Substitute the values $a = 4$ , $b = - 3$ , and $c = - n$ into the quadratic formula and solve for $m$ .

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

Step 1.4

Simplify.

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Raise $- 3$ to the power of $2$ .

$m = \frac{3 \pm \sqrt{9 - 4 \cdot 4 \cdot (- 1 n)}}{2 \cdot 4}$

Step 1.4.1.2

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

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Step 1.4.1.2.1

Multiply $- 4$ by $4$ .

$m = \frac{3 \pm \sqrt{9 - 16 \cdot (- 1 n)}}{2 \cdot 4}$

Step 1.4.1.2.2

Multiply $- 16$ by $- 1$ .

$m = \frac{3 \pm \sqrt{9 + 16 n}}{2 \cdot 4}$

Step 1.4.2

Multiply $2$ by $4$ .

$m = \frac{3 \pm \sqrt{9 + 16 n}}{8}$

Step 1.5

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

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

Raise $- 3$ to the power of $2$ .

$m = \frac{3 \pm \sqrt{9 - 4 \cdot 4 \cdot (- 1 n)}}{2 \cdot 4}$

Step 1.5.1.2

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

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Step 1.5.1.2.1

Multiply $- 4$ by $4$ .

$m = \frac{3 \pm \sqrt{9 - 16 \cdot (- 1 n)}}{2 \cdot 4}$

Step 1.5.1.2.2

Multiply $- 16$ by $- 1$ .

$m = \frac{3 \pm \sqrt{9 + 16 n}}{2 \cdot 4}$

Step 1.5.2

Multiply $2$ by $4$ .

$m = \frac{3 \pm \sqrt{9 + 16 n}}{8}$

Step 1.5.3

Change the $\pm$ to $+$ .

$m = \frac{3 + \sqrt{9 + 16 n}}{8}$

Step 1.6

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

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Step 1.6.1

Simplify the numerator.

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Step 1.6.1.1

Raise $- 3$ to the power of $2$ .

$m = \frac{3 \pm \sqrt{9 - 4 \cdot 4 \cdot (- 1 n)}}{2 \cdot 4}$

Step 1.6.1.2

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

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Step 1.6.1.2.1

Multiply $- 4$ by $4$ .

$m = \frac{3 \pm \sqrt{9 - 16 \cdot (- 1 n)}}{2 \cdot 4}$

Step 1.6.1.2.2

Multiply $- 16$ by $- 1$ .

$m = \frac{3 \pm \sqrt{9 + 16 n}}{2 \cdot 4}$

Step 1.6.2

Multiply $2$ by $4$ .

$m = \frac{3 \pm \sqrt{9 + 16 n}}{8}$

Step 1.6.3

Change the $\pm$ to $-$ .

$m = \frac{3 - \sqrt{9 + 16 n}}{8}$

Step 1.7

The final answer is the combination of both solutions.

$m = \frac{3 + \sqrt{9 + 16 n}}{8}, \frac{3 - \sqrt{9 + 16 n}}{8}$

Step 2

A linear equation is an equation of a straight line, which means that the degree of a linear equation must be $0$ or $1$ for each of its variables. In this case, the degree of the variable in the equation violates the linear equation definition, which means that the equation is not a linear equation.

Not Linear