Algebra Examples

$7 - 7 x = (3 x + 2) (x - 1)$

Step 1

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

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Step 1.1

Simplify the right side.

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Step 1.1.1

Simplify $(3 x + 2) (x - 1)$ .

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Step 1.1.1.1

Expand $(3 x + 2) (x - 1)$ using the FOIL Method.

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Step 1.1.1.1.1

Apply the distributive property.

$7 - 7 x = 3 x (x - 1) + 2 (x - 1)$

Step 1.1.1.1.2

Apply the distributive property.

$7 - 7 x = 3 x \cdot x + 3 x \cdot - 1 + 2 (x - 1)$

Step 1.1.1.1.3

Apply the distributive property.

$7 - 7 x = 3 x \cdot x + 3 x \cdot - 1 + 2 x + 2 \cdot - 1$

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 $x$ .

$7 - 7 x = 3 (x \cdot x) + 3 x \cdot - 1 + 2 x + 2 \cdot - 1$

Step 1.1.1.2.1.1.2

Multiply $x$ by $x$ .

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

Step 1.1.1.2.1.2

Multiply $- 1$ by $3$ .

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

Step 1.1.1.2.1.3

Multiply $2$ by $- 1$ .

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

Step 1.1.1.2.2

Add $- 3 x$ and $2 x$ .

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

Step 1.2

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

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Step 1.2.1

Subtract $3 x^{2}$ from both sides of the equation.

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

Step 1.2.2

Add $x$ to both sides of the equation.

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

Step 1.2.3

Add $2$ to both sides of the equation.

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

Step 1.3

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

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Step 1.3.1

Add $7$ and $2$ .

$- 7 x - 3 x^{2} + x + 9 = 0$

Step 1.3.2

Add $- 7 x$ and $x$ .

$- 3 x^{2} - 6 x + 9 = 0$

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 $a = - 3$ , $b = - 6$ , and $c = 9$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (- 3 \cdot 9)}}{2 \cdot - 3}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 3 \cdot 9}}{2 \cdot - 3}$

Step 4.1.2

Multiply $- 4 \cdot - 3 \cdot 9$ .

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Step 4.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{6 \pm \sqrt{36 + 12 \cdot 9}}{2 \cdot - 3}$

Step 4.1.2.2

Multiply $12$ by $9$ .

$x = \frac{6 \pm \sqrt{36 + 108}}{2 \cdot - 3}$

Step 4.1.3

Add $36$ and $108$ .

$x = \frac{6 \pm \sqrt{144}}{2 \cdot - 3}$

Step 4.1.4

Rewrite $144$ as $12^{2}$ .

$x = \frac{6 \pm \sqrt{12^{2}}}{2 \cdot - 3}$

Step 4.1.5

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

$x = \frac{6 \pm 12}{2 \cdot - 3}$

Step 4.2

Multiply $2$ by $- 3$ .

$x = \frac{6 \pm 12}{- 6}$

Step 4.3

Simplify $\frac{6 \pm 12}{- 6}$ .

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

Step 4.4

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

$x = - 1 \cdot (1 \pm 2)$

Step 4.5

Rewrite $- 1 \cdot (1 \pm 2)$ as $- (1 \pm 2)$ .

$x = - (1 \pm 2)$

Step 5

The final answer is the combination of both solutions.

$x = - 3, 1$