Algebra Examples

$- 3 x^{2} - 42 x - 51 = - 12 x$

Step 1

Add $12 x$ to both sides of the equation.

$- 3 x^{2} - 42 x - 51 + 12 x = 0$

Step 2

Add $- 42 x$ and $12 x$ .

$- 3 x^{2} - 30 x - 51 = 0$

Step 3

Factor $3$ out of $- 3 x^{2} - 30 x - 51$ .

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Step 3.1

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

$3 (- x^{2}) - 30 x - 51 = 0$

Step 3.2

Factor $3$ out of $- 30 x$ .

$3 (- x^{2}) + 3 (- 10 x) - 51 = 0$

Step 3.3

Factor $3$ out of $- 51$ .

$3 (- x^{2}) + 3 (- 10 x) + 3 (- 17) = 0$

Step 3.4

Factor $3$ out of $3 (- x^{2}) + 3 (- 10 x)$ .

$3 (- x^{2} - 10 x) + 3 (- 17) = 0$

Step 3.5

Factor $3$ out of $3 (- x^{2} - 10 x) + 3 (- 17)$ .

$3 (- x^{2} - 10 x - 17) = 0$

Step 4

Divide each term in $3 (- x^{2} - 10 x - 17) = 0$ by $3$ and simplify.

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Step 4.1

Divide each term in $3 (- x^{2} - 10 x - 17) = 0$ by $3$ .

$\frac{3 (- x^{2} - 10 x - 17)}{3} = \frac{0}{3}$

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.

$\frac{3 (- x^{2} - 10 x - 17)}{3} = \frac{0}{3}$

Step 4.2.1.2

Divide $- x^{2} - 10 x - 17$ by $1$ .

$- x^{2} - 10 x - 17 = \frac{0}{3}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $3$ .

$- x^{2} - 10 x - 17 = 0$

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 $a = - 1$ , $b = - 10$ , and $c = - 17$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (- 1 \cdot - 17)}}{2 \cdot - 1}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot - 1 \cdot - 17}}{2 \cdot - 1}$

Step 7.1.2

Multiply $- 4 \cdot - 1 \cdot - 17$ .

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Step 7.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{10 \pm \sqrt{100 + 4 \cdot - 17}}{2 \cdot - 1}$

Step 7.1.2.2

Multiply $4$ by $- 17$ .

$x = \frac{10 \pm \sqrt{100 - 68}}{2 \cdot - 1}$

Step 7.1.3

Subtract $68$ from $100$ .

$x = \frac{10 \pm \sqrt{32}}{2 \cdot - 1}$

Step 7.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 7.1.4.1

Factor $16$ out of $32$ .

$x = \frac{10 \pm \sqrt{16 (2)}}{2 \cdot - 1}$

Step 7.1.4.2

Rewrite $16$ as $4^{2}$ .

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

Step 7.1.5

Pull terms out from under the radical.

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

Step 7.2

Multiply $2$ by $- 1$ .

$x = \frac{10 \pm 4 \sqrt{2}}{- 2}$

Step 7.3

Simplify $\frac{10 \pm 4 \sqrt{2}}{- 2}$ .

$x = \frac{5 \pm 2 \sqrt{2}}{- 1}$

Step 7.4

Move the negative one from the denominator of $\frac{5 \pm 2 \sqrt{2}}{- 1}$ .

$x = - 1 \cdot (5 \pm 2 \sqrt{2})$

Step 7.5

Rewrite $- 1 \cdot (5 \pm 2 \sqrt{2})$ as $- (5 \pm 2 \sqrt{2})$ .

$x = - (5 \pm 2 \sqrt{2})$

Step 8

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot - 1 \cdot - 17}}{2 \cdot - 1}$

Step 8.1.2

Multiply $- 4 \cdot - 1 \cdot - 17$ .

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Step 8.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{10 \pm \sqrt{100 + 4 \cdot - 17}}{2 \cdot - 1}$

Step 8.1.2.2

Multiply $4$ by $- 17$ .

$x = \frac{10 \pm \sqrt{100 - 68}}{2 \cdot - 1}$

Step 8.1.3

Subtract $68$ from $100$ .

$x = \frac{10 \pm \sqrt{32}}{2 \cdot - 1}$

Step 8.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 8.1.4.1

Factor $16$ out of $32$ .

$x = \frac{10 \pm \sqrt{16 (2)}}{2 \cdot - 1}$

Step 8.1.4.2

Rewrite $16$ as $4^{2}$ .

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

Step 8.1.5

Pull terms out from under the radical.

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

Step 8.2

Multiply $2$ by $- 1$ .

$x = \frac{10 \pm 4 \sqrt{2}}{- 2}$

Step 8.3

Simplify $\frac{10 \pm 4 \sqrt{2}}{- 2}$ .

$x = \frac{5 \pm 2 \sqrt{2}}{- 1}$

Step 8.4

Move the negative one from the denominator of $\frac{5 \pm 2 \sqrt{2}}{- 1}$ .

$x = - 1 \cdot (5 \pm 2 \sqrt{2})$

Step 8.5

Rewrite $- 1 \cdot (5 \pm 2 \sqrt{2})$ as $- (5 \pm 2 \sqrt{2})$ .

$x = - (5 \pm 2 \sqrt{2})$

Step 8.6

Change the $\pm$ to $+$ .

$x = - (5 + 2 \sqrt{2})$

Step 8.7

Apply the distributive property.

$x = - 1 \cdot 5 - (2 \sqrt{2})$

Step 8.8

Multiply $- 1$ by $5$ .

$x = - 5 - (2 \sqrt{2})$

Step 8.9

Multiply $2$ by $- 1$ .

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

Step 9

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot - 1 \cdot - 17}}{2 \cdot - 1}$

Step 9.1.2

Multiply $- 4 \cdot - 1 \cdot - 17$ .

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Step 9.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{10 \pm \sqrt{100 + 4 \cdot - 17}}{2 \cdot - 1}$

Step 9.1.2.2

Multiply $4$ by $- 17$ .

$x = \frac{10 \pm \sqrt{100 - 68}}{2 \cdot - 1}$

Step 9.1.3

Subtract $68$ from $100$ .

$x = \frac{10 \pm \sqrt{32}}{2 \cdot - 1}$

Step 9.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 9.1.4.1

Factor $16$ out of $32$ .

$x = \frac{10 \pm \sqrt{16 (2)}}{2 \cdot - 1}$

Step 9.1.4.2

Rewrite $16$ as $4^{2}$ .

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

Step 9.1.5

Pull terms out from under the radical.

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

Step 9.2

Multiply $2$ by $- 1$ .

$x = \frac{10 \pm 4 \sqrt{2}}{- 2}$

Step 9.3

Simplify $\frac{10 \pm 4 \sqrt{2}}{- 2}$ .

$x = \frac{5 \pm 2 \sqrt{2}}{- 1}$

Step 9.4

Move the negative one from the denominator of $\frac{5 \pm 2 \sqrt{2}}{- 1}$ .

$x = - 1 \cdot (5 \pm 2 \sqrt{2})$

Step 9.5

Rewrite $- 1 \cdot (5 \pm 2 \sqrt{2})$ as $- (5 \pm 2 \sqrt{2})$ .

$x = - (5 \pm 2 \sqrt{2})$

Step 9.6

Change the $\pm$ to $-$ .

$x = - (5 - 2 \sqrt{2})$

Step 9.7

Apply the distributive property.

$x = - 1 \cdot 5 - (- 2 \sqrt{2})$

Step 9.8

Multiply $- 1$ by $5$ .

$x = - 5 - (- 2 \sqrt{2})$

Step 9.9

Multiply $- 2$ by $- 1$ .

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

Step 10

The final answer is the combination of both solutions.

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

Step 11

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 7.82842712 \dots, - 2.17157287 \dots$