Find the Equation Given the Roots 3/2 , 1/4

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

Step 1

Roots are the points where the graph intercepts with the x-axis $(y = 0)$ .

$y = 0$ at the roots

Step 2

The root at $x = \frac{3}{2}$ was found by solving for $x$ when $x - (\frac{3}{2}) = y$ and $y = 0$ .

The factor is $x - \frac{3}{2}$

Step 3

The root at $x = \frac{1}{4}$ was found by solving for $x$ when $x - (\frac{1}{4}) = y$ and $y = 0$ .

The factor is $x - \frac{1}{4}$

Step 4

Combine all the factors into a single equation.

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

Step 5

Multiply all the factors to simplify the equation $y = (x - \frac{3}{2}) (x - \frac{1}{4})$ .

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

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

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

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

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

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

Multiply $- 1$ by $- 1$ .

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

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

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

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

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

Multiply $2$ by $4$ .

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

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

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

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

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

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

Multiply $2$ by $2$ .

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

Combine the numerators over the common denominator.

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

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

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

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

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

Combine the numerators over the common denominator.

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

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

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

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

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

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

Multiply $4$ by $2$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Simplify each term.

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

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

Multiply $2$ by $- 3$ .

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

Subtract $6 x$ from $- x$ .

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

Apply the distributive property.

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

Multiply $2$ by $4$ .

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

Multiply $2$ by $- 7$ .

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

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

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

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

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

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $4$ by $2$ .

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

Multiply $- 3$ by $4$ .

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

Multiply $2$ by $- 1$ .

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

Multiply $- 1$ by $- 3$ .

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

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

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

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

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

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

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

Cancel the common factor of $8$ .

Cancel the common factor.

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

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

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

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

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

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

Cancel the common factors.

Factor $2$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Step 6

Find the Equation Given the Roots 3/2 , 1/4

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