Algebra Examples

$\frac{1}{7} x^{2} + \frac{1}{21} x - 1 = 0$

Step 1

Simplify the left side.

Tap for more steps...

Step 1.1

Simplify each term.

Tap for more steps...

Step 1.1.1

Combine $\frac{1}{7}$ and $x^{2}$ .

$\frac{x^{2}}{7} + \frac{1}{21} x - 1 = 0$

Step 1.1.2

Combine $\frac{1}{21}$ and $x$ .

$\frac{x^{2}}{7} + \frac{x}{21} - 1 = 0$

Step 2

Multiply through by the least common denominator $21$ , then simplify.

Tap for more steps...

Step 2.1

Apply the distributive property.

$21 (\frac{x^{2}}{7}) + 21 (\frac{x}{21}) + 21 \cdot - 1 = 0$

Step 2.2

Simplify.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $7$ .

Tap for more steps...

Step 2.2.1.1

Factor $7$ out of $21$ .

$7 (3) (\frac{x^{2}}{7}) + 21 (\frac{x}{21}) + 21 \cdot - 1 = 0$

Step 2.2.1.2

Cancel the common factor.

$7 \cdot (3 (\frac{x^{2}}{7})) + 21 (\frac{x}{21}) + 21 \cdot - 1 = 0$

Step 2.2.1.3

Rewrite the expression.

$3 x^{2} + 21 (\frac{x}{21}) + 21 \cdot - 1 = 0$

Step 2.2.2

Cancel the common factor of $21$ .

Tap for more steps...

Step 2.2.2.1

Cancel the common factor.

$3 x^{2} + 21 (\frac{x}{21}) + 21 \cdot - 1 = 0$

Step 2.2.2.2

Rewrite the expression.

$3 x^{2} + x + 21 \cdot - 1 = 0$

Step 2.2.3

Multiply $21$ by $- 1$ .

$3 x^{2} + x - 21 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 3$ , $b = 1$ , and $c = - 21$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (3 \cdot - 21)}}{2 \cdot 3}$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Simplify the numerator.

Tap for more steps...

Step 5.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 3 \cdot - 21}}{2 \cdot 3}$

Step 5.1.2

Multiply $- 4 \cdot 3 \cdot - 21$ .

Tap for more steps...

Step 5.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 1 \pm \sqrt{1 - 12 \cdot - 21}}{2 \cdot 3}$

Step 5.1.2.2

Multiply $- 12$ by $- 21$ .

$x = \frac{- 1 \pm \sqrt{1 + 252}}{2 \cdot 3}$

Step 5.1.3

Add $1$ and $252$ .

$x = \frac{- 1 \pm \sqrt{253}}{2 \cdot 3}$

Step 5.2

Multiply $2$ by $3$ .

$x = \frac{- 1 \pm \sqrt{253}}{6}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{- 1 \pm \sqrt{253}}{6}$

Decimal Form:

$x = 2.48432895 \dots, - 2.81766228 \dots$