Calculus Examples

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

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = - 2$ , $b = - x$ , and $c = x^{2}$ into the quadratic formula and solve for $m$ .

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

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Apply the product rule to $- x$ .

$m = \frac{x \pm \sqrt{{(- 1)}^{2} x^{2} - 4 \cdot - 2 \cdot x^{2}}}{2 \cdot - 2}$

Step 3.1.2

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

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

Step 3.1.3

Multiply $x^{2}$ by $1$ .

$m = \frac{x \pm \sqrt{x^{2} - 4 \cdot - 2 \cdot x^{2}}}{2 \cdot - 2}$

Step 3.1.4

Multiply $- 4$ by $- 2$ .

$m = \frac{x \pm \sqrt{x^{2} + 8 \cdot x^{2}}}{2 \cdot - 2}$

Step 3.1.5

Add $x^{2}$ and $8 x^{2}$ .

$m = \frac{x \pm \sqrt{9 x^{2}}}{2 \cdot - 2}$

Step 3.1.6

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

$m = \frac{x \pm \sqrt{{(3 x)}^{2}}}{2 \cdot - 2}$

Step 3.1.7

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

$m = \frac{x \pm 3 x}{2 \cdot - 2}$

Step 3.2

Multiply $2$ by $- 2$ .

$m = \frac{x \pm 3 x}{- 4}$

Step 3.3

Move the negative in front of the fraction.

$m = - \frac{x \pm 3 x}{4}$

Step 4

The final answer is the combination of both solutions.

$m = - x$

$m = \frac{x}{2}$