Algebra Examples

$3 x^{2} + 1 - 7 x = 13 x^{2}$

Step 1

Move all terms containing $x$ to the left side of the equation.

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Step 1.1

Subtract $13 x^{2}$ from both sides of the equation.

$3 x^{2} + 1 - 7 x - 13 x^{2} = 0$

Step 1.2

Subtract $13 x^{2}$ from $3 x^{2}$ .

$- 10 x^{2} + 1 - 7 x = 0$

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 $a = - 10$ , $b = - 7$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{7 \pm \sqrt{{(- 7)}^{2} - 4 \cdot (- 10 \cdot 1)}}{2 \cdot - 10}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 7$ to the power of $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot - 10 \cdot 1}}{2 \cdot - 10}$

Step 4.1.2

Multiply $- 4 \cdot - 10 \cdot 1$ .

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Step 4.1.2.1

Multiply $- 4$ by $- 10$ .

$x = \frac{7 \pm \sqrt{49 + 40 \cdot 1}}{2 \cdot - 10}$

Step 4.1.2.2

Multiply $40$ by $1$ .

$x = \frac{7 \pm \sqrt{49 + 40}}{2 \cdot - 10}$

Step 4.1.3

Add $49$ and $40$ .

$x = \frac{7 \pm \sqrt{89}}{2 \cdot - 10}$

Step 4.2

Multiply $2$ by $- 10$ .

$x = \frac{7 \pm \sqrt{89}}{- 20}$

Step 4.3

Move the negative in front of the fraction.

$x = - \frac{7 \pm \sqrt{89}}{20}$

Step 5

The final answer is the combination of both solutions.

$x = - \frac{7 + \sqrt{89}}{20}, - \frac{7 - \sqrt{89}}{20}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{7 + \sqrt{89}}{20}, - \frac{7 - \sqrt{89}}{20}$

Decimal Form:

$x = - 0.82169905 \dots, 0.12169905 \dots$