Find the Equation Given the Roots 3 , -11/4

$3$ , $- \frac{11}{4}$

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 $x = 3$ was found by solving for $x$ when $x - (3) = y$ and $y = 0$ .

The factor is $x - 3$

Step 3

The root at $x = - \frac{11}{4}$ was found by solving for $x$ when $x - (- \frac{11}{4}) = y$ and $y = 0$ .

The factor is $x + \frac{11}{4}$

Step 4

Combine all the factors into a single equation.

$y = (x - 3) (x + \frac{11}{4})$

Step 5

Multiply all the factors to simplify the equation $y = (x - 3) (x + \frac{11}{4})$ .

Expand $(x - 3) (x + \frac{11}{4})$ using the FOIL Method.

Apply the distributive property.

$y = x (x + \frac{11}{4}) - 3 (x + \frac{11}{4})$

Apply the distributive property.

$y = x \cdot x + x (\frac{11}{4}) - 3 (x + \frac{11}{4})$

Apply the distributive property.

$y = x \cdot x + x (\frac{11}{4}) - 3 x - 3 (\frac{11}{4})$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = x^{2} + x (\frac{11}{4}) - 3 x - 3 (\frac{11}{4})$

Combine $x$ and $\frac{11}{4}$ .

$y = x^{2} + \frac{x \cdot 11}{4} - 3 x - 3 (\frac{11}{4})$

Move $11$ to the left of $x$ .

$y = x^{2} + \frac{11 \cdot x}{4} - 3 x - 3 (\frac{11}{4})$

Multiply $- 3 (\frac{11}{4})$ .

Combine $- 3$ and $\frac{11}{4}$ .

$y = x^{2} + \frac{11 x}{4} - 3 x + \frac{- 3 \cdot 11}{4}$

Multiply $- 3$ by $11$ .

$y = x^{2} + \frac{11 x}{4} - 3 x + \frac{- 33}{4}$

Move the negative in front of the fraction.

$y = x^{2} + \frac{11 x}{4} - 3 x - \frac{33}{4}$

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

$y = x^{2} + \frac{11 x}{4} - 3 x \cdot \frac{4}{4} - \frac{33}{4}$

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

$y = x^{2} + \frac{11 x}{4} + \frac{- 3 x \cdot 4}{4} - \frac{33}{4}$

Combine the numerators over the common denominator.

$y = x^{2} + \frac{11 x - 3 x \cdot 4}{4} - \frac{33}{4}$

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

$y = x^{2} \cdot \frac{4}{4} + \frac{11 x - 3 x \cdot 4}{4} - \frac{33}{4}$

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

$y = \frac{x^{2} \cdot 4}{4} + \frac{11 x - 3 x \cdot 4}{4} - \frac{33}{4}$

Combine the numerators over the common denominator.

$y = \frac{x^{2} \cdot 4 + 11 x - 3 x \cdot 4}{4} - \frac{33}{4}$

Combine the numerators over the common denominator.

$y = \frac{x^{2} \cdot 4 + 11 x - 3 x \cdot 4 - 33}{4}$

Simplify the numerator.

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

$y = \frac{4 \cdot x^{2} + 11 x - 3 x \cdot 4 - 33}{4}$

Multiply $4$ by $- 3$ .

$y = \frac{4 x^{2} + 11 x - 12 x - 33}{4}$

Subtract $12 x$ from $11 x$ .

$y = \frac{4 x^{2} - x - 33}{4}$

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 $a \cdot c = 4 \cdot - 33 = - 132$ and whose sum is $b = - 1$ .

Factor $- 1$ out of $- x$ .

$y = \frac{4 x^{2} - (x) - 33}{4}$

Rewrite $- 1$ as $11$ plus $- 12$

$y = \frac{4 x^{2} + (11 - 12) x - 33}{4}$

Apply the distributive property.

$y = \frac{4 x^{2} + 11 x - 12 x - 33}{4}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$y = \frac{(4 x^{2} + 11 x) - 12 x - 33}{4}$

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

$y = \frac{x (4 x + 11) - 3 (4 x + 11)}{4}$

Factor the polynomial by factoring out the greatest common factor, $4 x + 11$ .

$y = \frac{(4 x + 11) (x - 3)}{4}$

Expand $(4 x + 11) (x - 3)$ using the FOIL Method.

Apply the distributive property.

$y = \frac{4 x (x - 3) + 11 (x - 3)}{4}$

Apply the distributive property.

$y = \frac{4 x \cdot x + 4 x \cdot - 3 + 11 (x - 3)}{4}$

Apply the distributive property.

$y = \frac{4 x \cdot x + 4 x \cdot - 3 + 11 x + 11 \cdot - 3}{4}$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$y = \frac{4 (x \cdot x) + 4 x \cdot - 3 + 11 x + 11 \cdot - 3}{4}$

Multiply $x$ by $x$ .

$y = \frac{4 x^{2} + 4 x \cdot - 3 + 11 x + 11 \cdot - 3}{4}$

Multiply $- 3$ by $4$ .

$y = \frac{4 x^{2} - 12 x + 11 x + 11 \cdot - 3}{4}$

Multiply $11$ by $- 3$ .

$y = \frac{4 x^{2} - 12 x + 11 x - 33}{4}$

Add $- 12 x$ and $11 x$ .

$y = \frac{4 x^{2} - x - 33}{4}$

Split the fraction $\frac{4 x^{2} - x - 33}{4}$ into two fractions.

$y = \frac{4 x^{2} - x}{4} + \frac{- 33}{4}$

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

$y = \frac{4 x^{2}}{4} + \frac{- x}{4} + \frac{- 33}{4}$

Cancel the common factor of $4$ .

Cancel the common factor.

$y = \frac{4 x^{2}}{4} + \frac{- x}{4} + \frac{- 33}{4}$

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

$y = x^{2} + \frac{- x}{4} + \frac{- 33}{4}$

Move the negative in front of the fraction.

$y = x^{2} - \frac{x}{4} + \frac{- 33}{4}$

Move the negative in front of the fraction.

$y = x^{2} - \frac{x}{4} - \frac{33}{4}$

Step 6

Find the Equation Given the Roots 3 , -11/4

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