Algebra Examples

$(- \frac{3}{4}, 2)$

Step 1

$x = - \frac{3}{4}$ and $x = 2$ are the two real distinct solutions for the quadratic equation, which means that $x + \frac{3}{4}$ and $x - 2$ are the factors of the quadratic equation.

$(x + \frac{3}{4}) (x - 2) = 0$

Step 2

Expand $(x + \frac{3}{4}) (x - 2)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$x (x - 2) + \frac{3}{4} \cdot (x - 2) = 0$

Step 2.2

Apply the distributive property.

$x \cdot x + x \cdot - 2 + \frac{3}{4} \cdot (x - 2) = 0$

Step 2.3

Apply the distributive property.

$x \cdot x + x \cdot - 2 + \frac{3}{4} x + \frac{3}{4} \cdot - 2 = 0$

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 $x$ .

$x^{2} + x \cdot - 2 + \frac{3}{4} x + \frac{3}{4} \cdot - 2 = 0$

Step 3.1.2

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

$x^{2} - 2 \cdot x + \frac{3}{4} x + \frac{3}{4} \cdot - 2 = 0$

Step 3.1.3

Combine $\frac{3}{4}$ and $x$ .

$x^{2} - 2 x + \frac{3 x}{4} + \frac{3}{4} \cdot - 2 = 0$

Step 3.1.4

Cancel the common factor of $2$ .

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Step 3.1.4.1

Factor $2$ out of $4$ .

$x^{2} - 2 x + \frac{3 x}{4} + \frac{3}{2 (2)} \cdot - 2 = 0$

Step 3.1.4.2

Factor $2$ out of $- 2$ .

$x^{2} - 2 x + \frac{3 x}{4} + \frac{3}{2 \cdot 2} \cdot (2 \cdot - 1) = 0$

Step 3.1.4.3

Cancel the common factor.

$x^{2} - 2 x + \frac{3 x}{4} + \frac{3}{2 \cdot 2} \cdot (2 \cdot - 1) = 0$

Step 3.1.4.4

Rewrite the expression.

$x^{2} - 2 x + \frac{3 x}{4} + \frac{3}{2} \cdot - 1 = 0$

Step 3.1.5

Combine $\frac{3}{2}$ and $- 1$ .

$x^{2} - 2 x + \frac{3 x}{4} + \frac{3 \cdot - 1}{2} = 0$

Step 3.1.6

Multiply $3$ by $- 1$ .

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

Step 3.1.7

Move the negative in front of the fraction.

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

Step 3.2

To write $- 2 x$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x^{2} - 2 x \cdot \frac{4}{4} + \frac{3 x}{4} - \frac{3}{2} = 0$

Step 3.3

Combine $- 2 x$ and $\frac{4}{4}$ .

$x^{2} + \frac{- 2 x \cdot 4}{4} + \frac{3 x}{4} - \frac{3}{2} = 0$

Step 3.4

Combine the numerators over the common denominator.

$x^{2} + \frac{- 2 x \cdot 4 + 3 x}{4} - \frac{3}{2} = 0$

Step 3.5

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x^{2} \cdot \frac{4}{4} + \frac{- 2 x \cdot 4 + 3 x}{4} - \frac{3}{2} = 0$

Step 3.6

Combine $x^{2}$ and $\frac{4}{4}$ .

$\frac{x^{2} \cdot 4}{4} + \frac{- 2 x \cdot 4 + 3 x}{4} - \frac{3}{2} = 0$

Step 3.7

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 4 - 2 x \cdot 4 + 3 x}{4} - \frac{3}{2} = 0$

Step 3.8

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

$\frac{x^{2} \cdot 4 - 2 x \cdot 4 + 3 x}{4} - \frac{3}{2} \cdot \frac{2}{2} = 0$

Step 3.9

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.9.1

Multiply $\frac{3}{2}$ by $\frac{2}{2}$ .

$\frac{x^{2} \cdot 4 - 2 x \cdot 4 + 3 x}{4} - \frac{3 \cdot 2}{2 \cdot 2} = 0$

Step 3.9.2

Multiply $2$ by $2$ .

$\frac{x^{2} \cdot 4 - 2 x \cdot 4 + 3 x}{4} - \frac{3 \cdot 2}{4} = 0$

Step 3.10

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 4 - 2 x \cdot 4 + 3 x - 3 \cdot 2}{4} = 0$

Step 4

Simplify the numerator.

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Step 4.1

Move $4$ to the left of $x^{2}$ .

$\frac{4 \cdot x^{2} - 2 x \cdot 4 + 3 x - 3 \cdot 2}{4} = 0$

Step 4.2

Multiply $4$ by $- 2$ .

$\frac{4 x^{2} - 8 x + 3 x - 3 \cdot 2}{4} = 0$

Step 4.3

Multiply $- 3$ by $2$ .

$\frac{4 x^{2} - 8 x + 3 x - 6}{4} = 0$

Step 4.4

Add $- 8 x$ and $3 x$ .

$\frac{4 x^{2} - 5 x - 6}{4} = 0$

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 $a \cdot c = 4 \cdot - 6 = - 24$ and whose sum is $b = - 5$ .

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Step 4.5.1.1

Factor $- 5$ out of $- 5 x$ .

$\frac{4 x^{2} - 5 x - 6}{4} = 0$

Step 4.5.1.2

Rewrite $- 5$ as $3$ plus $- 8$

$\frac{4 x^{2} + (3 - 8) x - 6}{4} = 0$

Step 4.5.1.3

Apply the distributive property.

$\frac{4 x^{2} + 3 x - 8 x - 6}{4} = 0$

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.

$\frac{(4 x^{2} + 3 x) - 8 x - 6}{4} = 0$

Step 4.5.2.2

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

$\frac{x (4 x + 3) - 2 (4 x + 3)}{4} = 0$

Step 4.5.3

Factor the polynomial by factoring out the greatest common factor, $4 x + 3$ .

$\frac{(4 x + 3) (x - 2)}{4} = 0$

Step 5

Expand $(4 x + 3) (x - 2)$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\frac{4 x (x - 2) + 3 (x - 2)}{4} = 0$

Step 5.2

Apply the distributive property.

$\frac{4 x \cdot x + 4 x \cdot - 2 + 3 (x - 2)}{4} = 0$

Step 5.3

Apply the distributive property.

$\frac{4 x \cdot x + 4 x \cdot - 2 + 3 x + 3 \cdot - 2}{4} = 0$

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 $x$ .

$\frac{4 (x \cdot x) + 4 x \cdot - 2 + 3 x + 3 \cdot - 2}{4} = 0$

Step 6.1.1.2

Multiply $x$ by $x$ .

$\frac{4 x^{2} + 4 x \cdot - 2 + 3 x + 3 \cdot - 2}{4} = 0$

Step 6.1.2

Multiply $- 2$ by $4$ .

$\frac{4 x^{2} - 8 x + 3 x + 3 \cdot - 2}{4} = 0$

Step 6.1.3

Multiply $3$ by $- 2$ .

$\frac{4 x^{2} - 8 x + 3 x - 6}{4} = 0$

Step 6.2

Add $- 8 x$ and $3 x$ .

$\frac{4 x^{2} - 5 x - 6}{4} = 0$

Step 7

Split the fraction $\frac{4 x^{2} - 5 x - 6}{4}$ into two fractions.

$\frac{4 x^{2} - 5 x}{4} + \frac{- 6}{4} = 0$

Step 8

Split the fraction $\frac{4 x^{2} - 5 x}{4}$ into two fractions.

$\frac{4 x^{2}}{4} + \frac{- 5 x}{4} + \frac{- 6}{4} = 0$

Step 9

Cancel the common factor of $4$ .

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Step 9.1

Cancel the common factor.

$\frac{4 x^{2}}{4} + \frac{- 5 x}{4} + \frac{- 6}{4} = 0$

Step 9.2

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

$x^{2} + \frac{- 5 x}{4} + \frac{- 6}{4} = 0$

Step 10

Move the negative in front of the fraction.

$x^{2} - \frac{5 x}{4} + \frac{- 6}{4} = 0$

Step 11

Cancel the common factor of $- 6$ and $4$ .

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Step 11.1

Factor $2$ out of $- 6$ .

$x^{2} - \frac{5 x}{4} + \frac{2 (- 3)}{4} = 0$

Step 11.2

Cancel the common factors.

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Step 11.2.1

Factor $2$ out of $4$ .

$x^{2} - \frac{5 x}{4} + \frac{2 \cdot - 3}{2 \cdot 2} = 0$

Step 11.2.2

Cancel the common factor.

$x^{2} - \frac{5 x}{4} + \frac{2 \cdot - 3}{2 \cdot 2} = 0$

Step 11.2.3

Rewrite the expression.

$x^{2} - \frac{5 x}{4} + \frac{- 3}{2} = 0$

Step 12

Move the negative in front of the fraction.

$x^{2} - \frac{5 x}{4} - \frac{3}{2} = 0$

Step 13

The standard quadratic equation using the given set of solutions ${- \frac{3}{4}, 2}$ is $y = x^{2} - \frac{5 x}{4} - \frac{3}{2}$ .

$y = x^{2} - \frac{5 x}{4} - \frac{3}{2}$

Step 14