Algebra Examples

$(3 x + 7) (x - 1) = 3 (x + 5) - 12$

Step 1

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

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Step 1.1

Simplify the left side.

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Step 1.1.1

Simplify $(3 x + 7) (x - 1)$ .

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Step 1.1.1.1

Expand $(3 x + 7) (x - 1)$ using the FOIL Method.

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Step 1.1.1.1.1

Apply the distributive property.

$3 x (x - 1) + 7 (x - 1) = 3 (x + 5) - 12$

Step 1.1.1.1.2

Apply the distributive property.

$3 x \cdot x + 3 x \cdot - 1 + 7 (x - 1) = 3 (x + 5) - 12$

Step 1.1.1.1.3

Apply the distributive property.

$3 x \cdot x + 3 x \cdot - 1 + 7 x + 7 \cdot - 1 = 3 (x + 5) - 12$

Step 1.1.1.2

Simplify and combine like terms.

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Step 1.1.1.2.1

Simplify each term.

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Step 1.1.1.2.1.1

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

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Step 1.1.1.2.1.1.1

Move $x$ .

$3 (x \cdot x) + 3 x \cdot - 1 + 7 x + 7 \cdot - 1 = 3 (x + 5) - 12$

Step 1.1.1.2.1.1.2

Multiply $x$ by $x$ .

$3 x^{2} + 3 x \cdot - 1 + 7 x + 7 \cdot - 1 = 3 (x + 5) - 12$

Step 1.1.1.2.1.2

Multiply $- 1$ by $3$ .

$3 x^{2} - 3 x + 7 x + 7 \cdot - 1 = 3 (x + 5) - 12$

Step 1.1.1.2.1.3

Multiply $7$ by $- 1$ .

$3 x^{2} - 3 x + 7 x - 7 = 3 (x + 5) - 12$

Step 1.1.1.2.2

Add $- 3 x$ and $7 x$ .

$3 x^{2} + 4 x - 7 = 3 (x + 5) - 12$

Step 1.2

Simplify the right side.

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Step 1.2.1

Simplify $3 (x + 5) - 12$ .

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Apply the distributive property.

$3 x^{2} + 4 x - 7 = 3 x + 3 \cdot 5 - 12$

Step 1.2.1.1.2

Multiply $3$ by $5$ .

$3 x^{2} + 4 x - 7 = 3 x + 15 - 12$

Step 1.2.1.2

Subtract $12$ from $15$ .

$3 x^{2} + 4 x - 7 = 3 x + 3$

Step 1.3

Move all the expressions to the left side of the equation.

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Step 1.3.1

Subtract $3 x$ from both sides of the equation.

$3 x^{2} + 4 x - 7 - 3 x = 3$

Step 1.3.2

Subtract $3$ from both sides of the equation.

$3 x^{2} + 4 x - 7 - 3 x - 3 = 0$

Step 1.4

Simplify $3 x^{2} + 4 x - 7 - 3 x - 3$ .

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Step 1.4.1

Subtract $3 x$ from $4 x$ .

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

Step 1.4.2

Subtract $3$ from $- 7$ .

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

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 $a = 3$ , $b = 1$ , and $c = - 10$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (3 \cdot - 10)}}{2 \cdot 3}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 3 \cdot - 10}}{2 \cdot 3}$

Step 4.1.2

Multiply $- 4 \cdot 3 \cdot - 10$ .

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Step 4.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 1 \pm \sqrt{1 - 12 \cdot - 10}}{2 \cdot 3}$

Step 4.1.2.2

Multiply $- 12$ by $- 10$ .

$x = \frac{- 1 \pm \sqrt{1 + 120}}{2 \cdot 3}$

Step 4.1.3

Add $1$ and $120$ .

$x = \frac{- 1 \pm \sqrt{121}}{2 \cdot 3}$

Step 4.1.4

Rewrite $121$ as $11^{2}$ .

$x = \frac{- 1 \pm \sqrt{11^{2}}}{2 \cdot 3}$

Step 4.1.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 1 \pm 11}{2 \cdot 3}$

Step 4.2

Multiply $2$ by $3$ .

$x = \frac{- 1 \pm 11}{6}$

Step 5

The final answer is the combination of both solutions.

$x = \frac{5}{3}, - 2$