Algebra Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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 $x$ .

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.1.4

Combine $\frac{1}{2}$ and $x$ .

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

Step 3.1.5

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

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Step 3.1.5.1

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

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

Step 3.1.5.2

Multiply $2$ by $4$ .

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

Step 3.2

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

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

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

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

Step 3.3.2

Multiply $2$ by $2$ .

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 4

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

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Step 4.1

Reorder $x$ and $2$ .

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

Step 4.2

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

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

Step 5

Move the negative in front of the fraction.

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

Step 6

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

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

Step 7