Algebra Examples

$- 3$ and $- \frac{2}{3}$

Step 1

$x = - 3$ and $x = - \frac{2}{3}$ are the two real distinct solutions for the quadratic equation, which means that $x + 3$ and $x + \frac{2}{3}$ are the factors of the quadratic equation.

$(x + 3) (x + \frac{2}{3}) = 0$

Step 2

Expand $(x + 3) (x + \frac{2}{3})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$x (x + \frac{2}{3}) + 3 (x + \frac{2}{3}) = 0$

Step 2.2

Apply the distributive property.

$x \cdot x + x (\frac{2}{3}) + 3 (x + \frac{2}{3}) = 0$

Step 2.3

Apply the distributive property.

$x \cdot x + x (\frac{2}{3}) + 3 x + 3 (\frac{2}{3}) = 0$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x$ by $x$ .

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

Step 3.1.2

Combine $x$ and $\frac{2}{3}$ .

$x^{2} + \frac{x \cdot 2}{3} + 3 x + 3 (\frac{2}{3}) = 0$

Step 3.1.3

Move $2$ to the left of $x$ .

$x^{2} + \frac{2 \cdot x}{3} + 3 x + 3 (\frac{2}{3}) = 0$

Step 3.1.4

Cancel the common factor of $3$ .

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Step 3.1.4.1

Cancel the common factor.

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

Step 3.1.4.2

Rewrite the expression.

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

Step 3.2

To write $3 x$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x^{2} + \frac{2 x}{3} + 3 x \cdot \frac{3}{3} + 2 = 0$

Step 3.3

Combine $3 x$ and $\frac{3}{3}$ .

$x^{2} + \frac{2 x}{3} + \frac{3 x \cdot 3}{3} + 2 = 0$

Step 3.4

Combine the numerators over the common denominator.

$x^{2} + \frac{2 x + 3 x \cdot 3}{3} + 2 = 0$

Step 3.5

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x^{2} \cdot \frac{3}{3} + \frac{2 x + 3 x \cdot 3}{3} + 2 = 0$

Step 3.6

Combine $x^{2}$ and $\frac{3}{3}$ .

$\frac{x^{2} \cdot 3}{3} + \frac{2 x + 3 x \cdot 3}{3} + 2 = 0$

Step 3.7

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 3 + 2 x + 3 x \cdot 3}{3} + 2 = 0$

Step 3.8

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{x^{2} \cdot 3 + 2 x + 3 x \cdot 3}{3} + 2 \cdot \frac{3}{3} = 0$

Step 3.9

Combine $2$ and $\frac{3}{3}$ .

$\frac{x^{2} \cdot 3 + 2 x + 3 x \cdot 3}{3} + \frac{2 \cdot 3}{3} = 0$

Step 3.10

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 3 + 2 x + 3 x \cdot 3 + 2 \cdot 3}{3} = 0$

Step 4

Simplify the numerator.

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Step 4.1

Move $3$ to the left of $x^{2}$ .

$\frac{3 \cdot x^{2} + 2 x + 3 x \cdot 3 + 2 \cdot 3}{3} = 0$

Step 4.2

Multiply $3$ by $3$ .

$\frac{3 x^{2} + 2 x + 9 x + 2 \cdot 3}{3} = 0$

Step 4.3

Multiply $2$ by $3$ .

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

Step 4.4

Add $2 x$ and $9 x$ .

$\frac{3 x^{2} + 11 x + 6}{3} = 0$

Step 4.5

Factor by grouping.

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Step 4.5.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 6 = 18$ and whose sum is $b = 11$ .

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Step 4.5.1.1

Factor $11$ out of $11 x$ .

$\frac{3 x^{2} + 11 (x) + 6}{3} = 0$

Step 4.5.1.2

Rewrite $11$ as $2$ plus $9$

$\frac{3 x^{2} + (2 + 9) x + 6}{3} = 0$

Step 4.5.1.3

Apply the distributive property.

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

Step 4.5.2

Factor out the greatest common factor from each group.

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Step 4.5.2.1

Group the first two terms and the last two terms.

$\frac{(3 x^{2} + 2 x) + 9 x + 6}{3} = 0$

Step 4.5.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{x (3 x + 2) + 3 (3 x + 2)}{3} = 0$

Step 4.5.3

Factor the polynomial by factoring out the greatest common factor, $3 x + 2$ .

$\frac{(3 x + 2) (x + 3)}{3} = 0$

Step 5

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

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Step 5.1

Apply the distributive property.

$\frac{3 x (x + 3) + 2 (x + 3)}{3} = 0$

Step 5.2

Apply the distributive property.

$\frac{3 x \cdot x + 3 x \cdot 3 + 2 (x + 3)}{3} = 0$

Step 5.3

Apply the distributive property.

$\frac{3 x \cdot x + 3 x \cdot 3 + 2 x + 2 \cdot 3}{3} = 0$

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

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

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Step 6.1.1.1

Move $x$ .

$\frac{3 (x \cdot x) + 3 x \cdot 3 + 2 x + 2 \cdot 3}{3} = 0$

Step 6.1.1.2

Multiply $x$ by $x$ .

$\frac{3 x^{2} + 3 x \cdot 3 + 2 x + 2 \cdot 3}{3} = 0$

Step 6.1.2

Multiply $3$ by $3$ .

$\frac{3 x^{2} + 9 x + 2 x + 2 \cdot 3}{3} = 0$

Step 6.1.3

Multiply $2$ by $3$ .

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

Step 6.2

Add $9 x$ and $2 x$ .

$\frac{3 x^{2} + 11 x + 6}{3} = 0$

Step 7

Split the fraction $\frac{3 x^{2} + 11 x + 6}{3}$ into two fractions.

$\frac{3 x^{2} + 11 x}{3} + \frac{6}{3} = 0$

Step 8

Split the fraction $\frac{3 x^{2} + 11 x}{3}$ into two fractions.

$\frac{3 x^{2}}{3} + \frac{11 x}{3} + \frac{6}{3} = 0$

Step 9

Cancel the common factor of $3$ .

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Step 9.1

Cancel the common factor.

$\frac{3 x^{2}}{3} + \frac{11 x}{3} + \frac{6}{3} = 0$

Step 9.2

Divide $x^{2}$ by $1$ .

$x^{2} + \frac{11 x}{3} + \frac{6}{3} = 0$

Step 10

Divide $6$ by $3$ .

$x^{2} + \frac{11 x}{3} + 2 = 0$

Step 11

The standard quadratic equation using the given set of solutions ${- 3, - \frac{2}{3}}$ is $y = x^{2} + \frac{11 x}{3} + 2$ .

$y = x^{2} + \frac{11 x}{3} + 2$

Step 12