Algebra Examples

$3 {(3 x + 1)}^{2} = 39$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Subtract $39$ from both sides of the equation.

$3 {(3 x + 1)}^{2} - 39 = 0$

Step 1.2

Simplify $3 {(3 x + 1)}^{2} - 39$ .

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Rewrite ${(3 x + 1)}^{2}$ as $(3 x + 1) (3 x + 1)$ .

$3 ((3 x + 1) (3 x + 1)) - 39 = 0$

Step 1.2.1.2

Expand $(3 x + 1) (3 x + 1)$ using the FOIL Method.

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Step 1.2.1.2.1

Apply the distributive property.

$3 (3 x (3 x + 1) + 1 (3 x + 1)) - 39 = 0$

Step 1.2.1.2.2

Apply the distributive property.

$3 (3 x (3 x) + 3 x \cdot 1 + 1 (3 x + 1)) - 39 = 0$

Step 1.2.1.2.3

Apply the distributive property.

$3 (3 x (3 x) + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1) - 39 = 0$

Step 1.2.1.3

Simplify and combine like terms.

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Step 1.2.1.3.1

Simplify each term.

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Step 1.2.1.3.1.1

Rewrite using the commutative property of multiplication.

$3 (3 \cdot 3 x \cdot x + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1) - 39 = 0$

Step 1.2.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.2.1.3.1.2.1

Move $x$ .

$3 (3 \cdot 3 (x \cdot x) + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1) - 39 = 0$

Step 1.2.1.3.1.2.2

Multiply $x$ by $x$ .

$3 (3 \cdot 3 x^{2} + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1) - 39 = 0$

Step 1.2.1.3.1.3

Multiply $3$ by $3$ .

$3 (9 x^{2} + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1) - 39 = 0$

Step 1.2.1.3.1.4

Multiply $3$ by $1$ .

$3 (9 x^{2} + 3 x + 1 (3 x) + 1 \cdot 1) - 39 = 0$

Step 1.2.1.3.1.5

Multiply $3 x$ by $1$ .

$3 (9 x^{2} + 3 x + 3 x + 1 \cdot 1) - 39 = 0$

Step 1.2.1.3.1.6

Multiply $1$ by $1$ .

$3 (9 x^{2} + 3 x + 3 x + 1) - 39 = 0$

Step 1.2.1.3.2

Add $3 x$ and $3 x$ .

$3 (9 x^{2} + 6 x + 1) - 39 = 0$

Step 1.2.1.4

Apply the distributive property.

$3 (9 x^{2}) + 3 (6 x) + 3 \cdot 1 - 39 = 0$

Step 1.2.1.5

Simplify.

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Step 1.2.1.5.1

Multiply $9$ by $3$ .

$27 x^{2} + 3 (6 x) + 3 \cdot 1 - 39 = 0$

Step 1.2.1.5.2

Multiply $6$ by $3$ .

$27 x^{2} + 18 x + 3 \cdot 1 - 39 = 0$

Step 1.2.1.5.3

Multiply $3$ by $1$ .

$27 x^{2} + 18 x + 3 - 39 = 0$

Step 1.2.2

Subtract $39$ from $3$ .

$27 x^{2} + 18 x - 36 = 0$

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 $a = 27$ , $b = 18$ , and $c = - 36$ into the quadratic formula and solve for $x$ .

$\frac{- 18 \pm \sqrt{18^{2} - 4 \cdot (27 \cdot - 36)}}{2 \cdot 27}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $18$ to the power of $2$ .

$x = \frac{- 18 \pm \sqrt{324 - 4 \cdot 27 \cdot - 36}}{2 \cdot 27}$

Step 4.1.2

Multiply $- 4 \cdot 27 \cdot - 36$ .

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Step 4.1.2.1

Multiply $- 4$ by $27$ .

$x = \frac{- 18 \pm \sqrt{324 - 108 \cdot - 36}}{2 \cdot 27}$

Step 4.1.2.2

Multiply $- 108$ by $- 36$ .

$x = \frac{- 18 \pm \sqrt{324 + 3888}}{2 \cdot 27}$

Step 4.1.3

Add $324$ and $3888$ .

$x = \frac{- 18 \pm \sqrt{4212}}{2 \cdot 27}$

Step 4.1.4

Rewrite $4212$ as $18^{2} \cdot 13$ .

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Step 4.1.4.1

Factor $324$ out of $4212$ .

$x = \frac{- 18 \pm \sqrt{324 (13)}}{2 \cdot 27}$

Step 4.1.4.2

Rewrite $324$ as $18^{2}$ .

$x = \frac{- 18 \pm \sqrt{18^{2} \cdot 13}}{2 \cdot 27}$

Step 4.1.5

Pull terms out from under the radical.

$x = \frac{- 18 \pm 18 \sqrt{13}}{2 \cdot 27}$

Step 4.2

Multiply $2$ by $27$ .

$x = \frac{- 18 \pm 18 \sqrt{13}}{54}$

Step 4.3

Simplify $\frac{- 18 \pm 18 \sqrt{13}}{54}$ .

$x = \frac{- 1 \pm \sqrt{13}}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{- 1 \pm \sqrt{13}}{3}$

Decimal Form:

$x = 0.86851709 \dots, - 1.53518375 \dots$