Algebra Examples

$3 x^{2} = 3 - 42 x$

Step 1

Add $42 x$ to both sides of the equation.

$3 x^{2} + 42 x = 3$

Step 2

Subtract $3$ from both sides of the equation.

$3 x^{2} + 42 x - 3 = 0$

Step 3

Factor $3$ out of $3 x^{2} + 42 x - 3$ .

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Step 3.1

Factor $3$ out of $3 x^{2}$ .

$3 (x^{2}) + 42 x - 3 = 0$

Step 3.2

Factor $3$ out of $42 x$ .

$3 (x^{2}) + 3 (14 x) - 3 = 0$

Step 3.3

Factor $3$ out of $- 3$ .

$3 (x^{2}) + 3 (14 x) + 3 (- 1) = 0$

Step 3.4

Factor $3$ out of $3 (x^{2}) + 3 (14 x)$ .

$3 (x^{2} + 14 x) + 3 (- 1) = 0$

Step 3.5

Factor $3$ out of $3 (x^{2} + 14 x) + 3 (- 1)$ .

$3 (x^{2} + 14 x - 1) = 0$

Step 4

Divide each term in $3 (x^{2} + 14 x - 1) = 0$ by $3$ and simplify.

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Step 4.1

Divide each term in $3 (x^{2} + 14 x - 1) = 0$ by $3$ .

$\frac{3 (x^{2} + 14 x - 1)}{3} = \frac{0}{3}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $3$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{3 (x^{2} + 14 x - 1)}{3} = \frac{0}{3}$

Step 4.2.1.2

Divide $x^{2} + 14 x - 1$ by $1$ .

$x^{2} + 14 x - 1 = \frac{0}{3}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $3$ .

$x^{2} + 14 x - 1 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 1$ , $b = 14$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{- 14 \pm \sqrt{14^{2} - 4 \cdot (1 \cdot - 1)}}{2 \cdot 1}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $14$ to the power of $2$ .

$x = \frac{- 14 \pm \sqrt{196 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 7.1.2

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

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 14 \pm \sqrt{196 - 4 \cdot - 1}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{- 14 \pm \sqrt{196 + 4}}{2 \cdot 1}$

Step 7.1.3

Add $196$ and $4$ .

$x = \frac{- 14 \pm \sqrt{200}}{2 \cdot 1}$

Step 7.1.4

Rewrite $200$ as $10^{2} \cdot 2$ .

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Step 7.1.4.1

Factor $100$ out of $200$ .

$x = \frac{- 14 \pm \sqrt{100 (2)}}{2 \cdot 1}$

Step 7.1.4.2

Rewrite $100$ as $10^{2}$ .

$x = \frac{- 14 \pm \sqrt{10^{2} \cdot 2}}{2 \cdot 1}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{- 14 \pm 10 \sqrt{2}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$x = \frac{- 14 \pm 10 \sqrt{2}}{2}$

Step 7.3

Simplify $\frac{- 14 \pm 10 \sqrt{2}}{2}$ .

$x = - 7 \pm 5 \sqrt{2}$

Step 8

The final answer is the combination of both solutions.

$x = - 7 + 5 \sqrt{2}, - 7 - 5 \sqrt{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - 7 + 5 \sqrt{2}, - 7 - 5 \sqrt{2}$

Decimal Form:

$x = 0.07106781 \dots, - 14.07106781 \dots$