Algebra Examples

$\frac{3}{2}$ , $\frac{1}{4}$

Step 1

$x = \frac{3}{2}$ and $x = \frac{1}{4}$ are the two real distinct solutions for the quadratic equation, which means that $x - \frac{3}{2}$ and $x - \frac{1}{4}$ are the factors of the quadratic equation.

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

Step 2

Expand $(x - \frac{3}{2}) (x - \frac{1}{4})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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 $x$ .

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

Step 3.1.2

Combine $x$ and $\frac{1}{4}$ .

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

Step 3.1.3

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

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

Step 3.1.4

Move $3$ to the left of $x$ .

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

Step 3.1.5

Multiply $- \frac{3}{2} (- \frac{1}{4})$ .

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Step 3.1.5.1

Multiply $- 1$ by $- 1$ .

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

Step 3.1.5.2

Multiply $\frac{3}{2}$ by $1$ .

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

Step 3.1.5.3

Multiply $\frac{3}{2}$ by $\frac{1}{4}$ .

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

Step 3.1.5.4

Multiply $2$ by $4$ .

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Multiply $2$ by $2$ .

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Combine the numerators over the common denominator.

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

Step 3.8

To write $\frac{x^{2} \cdot 4 - x - 3 x \cdot 2}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.9

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.9.1

Multiply $\frac{x^{2} \cdot 4 - x - 3 x \cdot 2}{4}$ by $\frac{2}{2}$ .

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

Step 3.9.2

Multiply $4$ by $2$ .

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

Step 3.10

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

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

Step 4.1.2

Multiply $2$ by $- 3$ .

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

Step 4.2

Subtract $6 x$ from $- x$ .

$\frac{(4 x^{2} - 7 x) \cdot 2 + 3}{8} = 0$

Step 4.3

Apply the distributive property.

$\frac{4 x^{2} \cdot 2 - 7 x \cdot 2 + 3}{8} = 0$

Step 4.4

Multiply $2$ by $4$ .

$\frac{8 x^{2} - 7 x \cdot 2 + 3}{8} = 0$

Step 4.5

Multiply $2$ by $- 7$ .

$\frac{8 x^{2} - 14 x + 3}{8} = 0$

Step 4.6

Factor by grouping.

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Step 4.6.1

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 $a \cdot c = 8 \cdot 3 = 24$ and whose sum is $b = - 14$ .

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Step 4.6.1.1

Factor $- 14$ out of $- 14 x$ .

$\frac{8 x^{2} - 14 x + 3}{8} = 0$

Step 4.6.1.2

Rewrite $- 14$ as $- 2$ plus $- 12$

$\frac{8 x^{2} + (- 2 - 12) x + 3}{8} = 0$

Step 4.6.1.3

Apply the distributive property.

$\frac{8 x^{2} - 2 x - 12 x + 3}{8} = 0$

Step 4.6.2

Factor out the greatest common factor from each group.

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Step 4.6.2.1

Group the first two terms and the last two terms.

$\frac{(8 x^{2} - 2 x) - 12 x + 3}{8} = 0$

Step 4.6.2.2

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

$\frac{2 x (4 x - 1) - 3 (4 x - 1)}{8} = 0$

Step 4.6.3

Factor the polynomial by factoring out the greatest common factor, $4 x - 1$ .

$\frac{(4 x - 1) (2 x - 3)}{8} = 0$

Step 5

Expand $(4 x - 1) (2 x - 3)$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\frac{4 x (2 x - 3) - 1 (2 x - 3)}{8} = 0$

Step 5.2

Apply the distributive property.

$\frac{4 x (2 x) + 4 x \cdot - 3 - 1 (2 x - 3)}{8} = 0$

Step 5.3

Apply the distributive property.

$\frac{4 x (2 x) + 4 x \cdot - 3 - 1 (2 x) - 1 \cdot - 3}{8} = 0$

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

Rewrite using the commutative property of multiplication.

$\frac{4 \cdot (2 x \cdot x) + 4 x \cdot - 3 - 1 (2 x) - 1 \cdot - 3}{8} = 0$

Step 6.1.2

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

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Step 6.1.2.1

Move $x$ .

$\frac{4 \cdot (2 (x \cdot x)) + 4 x \cdot - 3 - 1 (2 x) - 1 \cdot - 3}{8} = 0$

Step 6.1.2.2

Multiply $x$ by $x$ .

$\frac{4 \cdot (2 x^{2}) + 4 x \cdot - 3 - 1 (2 x) - 1 \cdot - 3}{8} = 0$

Step 6.1.3

Multiply $4$ by $2$ .

$\frac{8 x^{2} + 4 x \cdot - 3 - 1 (2 x) - 1 \cdot - 3}{8} = 0$

Step 6.1.4

Multiply $- 3$ by $4$ .

$\frac{8 x^{2} - 12 x - 1 (2 x) - 1 \cdot - 3}{8} = 0$

Step 6.1.5

Multiply $2$ by $- 1$ .

$\frac{8 x^{2} - 12 x - 2 x - 1 \cdot - 3}{8} = 0$

Step 6.1.6

Multiply $- 1$ by $- 3$ .

$\frac{8 x^{2} - 12 x - 2 x + 3}{8} = 0$

Step 6.2

Subtract $2 x$ from $- 12 x$ .

$\frac{8 x^{2} - 14 x + 3}{8} = 0$

Step 7

Split the fraction $\frac{8 x^{2} - 14 x + 3}{8}$ into two fractions.

$\frac{8 x^{2} - 14 x}{8} + \frac{3}{8} = 0$

Step 8

Split the fraction $\frac{8 x^{2} - 14 x}{8}$ into two fractions.

$\frac{8 x^{2}}{8} + \frac{- 14 x}{8} + \frac{3}{8} = 0$

Step 9

Cancel the common factor of $8$ .

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Step 9.1

Cancel the common factor.

$\frac{8 x^{2}}{8} + \frac{- 14 x}{8} + \frac{3}{8} = 0$

Step 9.2

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

$x^{2} + \frac{- 14 x}{8} + \frac{3}{8} = 0$

Step 10

Cancel the common factor of $- 14$ and $8$ .

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Step 10.1

Factor $2$ out of $- 14 x$ .

$x^{2} + \frac{2 (- 7 x)}{8} + \frac{3}{8} = 0$

Step 10.2

Cancel the common factors.

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Step 10.2.1

Factor $2$ out of $8$ .

$x^{2} + \frac{2 (- 7 x)}{2 (4)} + \frac{3}{8} = 0$

Step 10.2.2

Cancel the common factor.

$x^{2} + \frac{2 (- 7 x)}{2 \cdot 4} + \frac{3}{8} = 0$

Step 10.2.3

Rewrite the expression.

$x^{2} + \frac{- 7 x}{4} + \frac{3}{8} = 0$

Step 11

Move the negative in front of the fraction.

$x^{2} - \frac{7 x}{4} + \frac{3}{8} = 0$

Step 12

The standard quadratic equation using the given set of solutions ${\frac{3}{2}, \frac{1}{4}}$ is $y = x^{2} - \frac{7 x}{4} + \frac{3}{8}$ .

$y = x^{2} - \frac{7 x}{4} + \frac{3}{8}$

Step 13