Find the Quadratic Equation {3/4,1/8}

${\frac{3}{4}, \frac{1}{8}}$

Step 1

$x = \frac{3}{4}$ and $x = \frac{1}{8}$ are the two real distinct solutions for the quadratic equation, which means that $x - \frac{3}{4}$ and $x - \frac{1}{8}$ are the factors of the quadratic equation.

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

Step 2

Expand $(x - \frac{3}{4}) (x - \frac{1}{8})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Combine $x$ and $\frac{1}{8}$ .

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

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

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

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

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

Multiply $- \frac{3}{4} (- \frac{1}{8})$ .

Multiply $- 1$ by $- 1$ .

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

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

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

Multiply $\frac{3}{4}$ by $\frac{1}{8}$ .

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

Multiply $4$ by $8$ .

$x^{2} - \frac{x}{8} - \frac{3 x}{4} + \frac{3}{32} = 0$

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

$x^{2} - \frac{x}{8} - \frac{3 x}{4} \cdot \frac{2}{2} + \frac{3}{32} = 0$

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

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

$x^{2} - \frac{x}{8} - \frac{3 x \cdot 2}{4 \cdot 2} + \frac{3}{32} = 0$

Multiply $4$ by $2$ .

$x^{2} - \frac{x}{8} - \frac{3 x \cdot 2}{8} + \frac{3}{32} = 0$

Combine the numerators over the common denominator.

$x^{2} + \frac{- x - 3 x \cdot 2}{8} + \frac{3}{32} = 0$

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$x^{2} \cdot \frac{8}{8} + \frac{- x - 3 x \cdot 2}{8} + \frac{3}{32} = 0$

Combine $x^{2}$ and $\frac{8}{8}$ .

$\frac{x^{2} \cdot 8}{8} + \frac{- x - 3 x \cdot 2}{8} + \frac{3}{32} = 0$

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 8 - x - 3 x \cdot 2}{8} + \frac{3}{32} = 0$

To write $\frac{x^{2} \cdot 8 - x - 3 x \cdot 2}{8}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{x^{2} \cdot 8 - x - 3 x \cdot 2}{8} \cdot \frac{4}{4} + \frac{3}{32} = 0$

Write each expression with a common denominator of $32$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x^{2} \cdot 8 - x - 3 x \cdot 2}{8}$ by $\frac{4}{4}$ .

$\frac{(x^{2} \cdot 8 - x - 3 x \cdot 2) \cdot 4}{8 \cdot 4} + \frac{3}{32} = 0$

Multiply $8$ by $4$ .

$\frac{(x^{2} \cdot 8 - x - 3 x \cdot 2) \cdot 4}{32} + \frac{3}{32} = 0$

Combine the numerators over the common denominator.

$\frac{(x^{2} \cdot 8 - x - 3 x \cdot 2) \cdot 4 + 3}{32} = 0$

Step 4

Simplify the numerator.

Simplify each term.

Move $8$ to the left of $x^{2}$ .

$\frac{(8 \cdot x^{2} - x - 3 x \cdot 2) \cdot 4 + 3}{32} = 0$

Multiply $2$ by $- 3$ .

$\frac{(8 x^{2} - x - 6 x) \cdot 4 + 3}{32} = 0$

Subtract $6 x$ from $- x$ .

$\frac{(8 x^{2} - 7 x) \cdot 4 + 3}{32} = 0$

Apply the distributive property.

$\frac{8 x^{2} \cdot 4 - 7 x \cdot 4 + 3}{32} = 0$

Multiply $4$ by $8$ .

$\frac{32 x^{2} - 7 x \cdot 4 + 3}{32} = 0$

Multiply $4$ by $- 7$ .

$\frac{32 x^{2} - 28 x + 3}{32} = 0$

Factor by grouping.

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 $a \cdot c = 32 \cdot 3 = 96$ and whose sum is $b = - 28$ .

Factor $- 28$ out of $- 28 x$ .

$\frac{32 x^{2} - 28 x + 3}{32} = 0$

Rewrite $- 28$ as $- 4$ plus $- 24$

$\frac{32 x^{2} + (- 4 - 24) x + 3}{32} = 0$

Apply the distributive property.

$\frac{32 x^{2} - 4 x - 24 x + 3}{32} = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(32 x^{2} - 4 x) - 24 x + 3}{32} = 0$

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

$\frac{4 x (8 x - 1) - 3 (8 x - 1)}{32} = 0$

Factor the polynomial by factoring out the greatest common factor, $8 x - 1$ .

$\frac{(8 x - 1) (4 x - 3)}{32} = 0$

Step 5

Expand $(8 x - 1) (4 x - 3)$ using the FOIL Method.

Apply the distributive property.

$\frac{8 x (4 x - 3) - 1 (4 x - 3)}{32} = 0$

Apply the distributive property.

$\frac{8 x (4 x) + 8 x \cdot - 3 - 1 (4 x - 3)}{32} = 0$

Apply the distributive property.

$\frac{8 x (4 x) + 8 x \cdot - 3 - 1 (4 x) - 1 \cdot - 3}{32} = 0$

Step 6

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$\frac{8 \cdot (4 x \cdot x) + 8 x \cdot - 3 - 1 (4 x) - 1 \cdot - 3}{32} = 0$

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

Move $x$ .

$\frac{8 \cdot (4 (x \cdot x)) + 8 x \cdot - 3 - 1 (4 x) - 1 \cdot - 3}{32} = 0$

Multiply $x$ by $x$ .

$\frac{8 \cdot (4 x^{2}) + 8 x \cdot - 3 - 1 (4 x) - 1 \cdot - 3}{32} = 0$

Multiply $8$ by $4$ .

$\frac{32 x^{2} + 8 x \cdot - 3 - 1 (4 x) - 1 \cdot - 3}{32} = 0$

Multiply $- 3$ by $8$ .

$\frac{32 x^{2} - 24 x - 1 (4 x) - 1 \cdot - 3}{32} = 0$

Multiply $4$ by $- 1$ .

$\frac{32 x^{2} - 24 x - 4 x - 1 \cdot - 3}{32} = 0$

Multiply $- 1$ by $- 3$ .

$\frac{32 x^{2} - 24 x - 4 x + 3}{32} = 0$

Subtract $4 x$ from $- 24 x$ .

$\frac{32 x^{2} - 28 x + 3}{32} = 0$

Step 7

Split the fraction $\frac{32 x^{2} - 28 x + 3}{32}$ into two fractions.

$\frac{32 x^{2} - 28 x}{32} + \frac{3}{32} = 0$

Step 8

Split the fraction $\frac{32 x^{2} - 28 x}{32}$ into two fractions.

$\frac{32 x^{2}}{32} + \frac{- 28 x}{32} + \frac{3}{32} = 0$

Step 9

Cancel the common factor of $32$ .

Cancel the common factor.

$\frac{32 x^{2}}{32} + \frac{- 28 x}{32} + \frac{3}{32} = 0$

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

$x^{2} + \frac{- 28 x}{32} + \frac{3}{32} = 0$

Step 10

Cancel the common factor of $- 28$ and $32$ .

Factor $4$ out of $- 28 x$ .

$x^{2} + \frac{4 (- 7 x)}{32} + \frac{3}{32} = 0$

Cancel the common factors.

Factor $4$ out of $32$ .

$x^{2} + \frac{4 (- 7 x)}{4 (8)} + \frac{3}{32} = 0$

Cancel the common factor.

$x^{2} + \frac{4 (- 7 x)}{4 \cdot 8} + \frac{3}{32} = 0$

Rewrite the expression.

$x^{2} + \frac{- 7 x}{8} + \frac{3}{32} = 0$

Step 11

Move the negative in front of the fraction.

$x^{2} - \frac{7 x}{8} + \frac{3}{32} = 0$

Step 12

The standard quadratic equation using the given set of solutions ${\frac{3}{4}, \frac{1}{8}}$ is $y = x^{2} - \frac{7 x}{8} + \frac{3}{32}$ .

$y = x^{2} - \frac{7 x}{8} + \frac{3}{32}$

Step 13

Find the Quadratic Equation {3/4,1/8}

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