Basic Math Examples

$3 x^{2} - 2 x + k = 0$

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 $a = 3$ , $b = - 2$ , and $c = k$ into the quadratic formula and solve for $x$ .

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

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 3 \cdot k}}{2 \cdot 3}$

Step 3.1.2

Multiply $- 4$ by $3$ .

$x = \frac{2 \pm \sqrt{4 - 12 \cdot k}}{2 \cdot 3}$

Step 3.1.3

Factor $4$ out of $4 - 12 k$ .

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Step 3.1.3.1

Factor $4$ out of $4$ .

$x = \frac{2 \pm \sqrt{4 (1) - 12 k}}{2 \cdot 3}$

Step 3.1.3.2

Factor $4$ out of $- 12 k$ .

$x = \frac{2 \pm \sqrt{4 (1) + 4 (- 3 k)}}{2 \cdot 3}$

Step 3.1.3.3

Factor $4$ out of $4 (1) + 4 (- 3 k)$ .

$x = \frac{2 \pm \sqrt{4 (1 - 3 k)}}{2 \cdot 3}$

Step 3.1.4

Rewrite $4 (1 - 3 k)$ as $2^{2} (1^{2} - 3 k)$ .

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Step 3.1.4.1

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

$x = \frac{2 \pm \sqrt{2^{2} (1 - 3 k)}}{2 \cdot 3}$

Step 3.1.4.2

Rewrite $1$ as $1^{2}$ .

$x = \frac{2 \pm \sqrt{2^{2} (1^{2} - 3 k)}}{2 \cdot 3}$

Step 3.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{1^{2} - 3 k}}{2 \cdot 3}$

Step 3.1.6

One to any power is one.

$x = \frac{2 \pm 2 \sqrt{1 - 3 k}}{2 \cdot 3}$

Step 3.2

Multiply $2$ by $3$ .

$x = \frac{2 \pm 2 \sqrt{1 - 3 k}}{6}$

Step 3.3

Simplify $\frac{2 \pm 2 \sqrt{1 - 3 k}}{6}$ .

$x = \frac{1 \pm \sqrt{1 - 3 k}}{3}$

Step 4

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{1 - 3 k}}{3}$

$x = \frac{1 - \sqrt{1 - 3 k}}{3}$