Find the Equation Given the Roots -1/6 , 4

$- \frac{1}{6}$ , $4$

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 $x = - \frac{1}{6}$ was found by solving for $x$ when $x - (- \frac{1}{6}) = y$ and $y = 0$ .

The factor is $x + \frac{1}{6}$

Step 3

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

The factor is $x - 4$

Step 4

Combine all the factors into a single equation.

$y = (x + \frac{1}{6}) (x - 4)$

Step 5

Multiply all the factors to simplify the equation $y = (x + \frac{1}{6}) (x - 4)$ .

Expand $(x + \frac{1}{6}) (x - 4)$ using the FOIL Method.

Apply the distributive property.

$y = x (x - 4) + \frac{1}{6} \cdot (x - 4)$

Apply the distributive property.

$y = x \cdot x + x \cdot - 4 + \frac{1}{6} \cdot (x - 4)$

Apply the distributive property.

$y = x \cdot x + x \cdot - 4 + \frac{1}{6} x + \frac{1}{6} \cdot - 4$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = x^{2} + x \cdot - 4 + \frac{1}{6} x + \frac{1}{6} \cdot - 4$

Move $- 4$ to the left of $x$ .

$y = x^{2} - 4 \cdot x + \frac{1}{6} x + \frac{1}{6} \cdot - 4$

Combine $\frac{1}{6}$ and $x$ .

$y = x^{2} - 4 x + \frac{x}{6} + \frac{1}{6} \cdot - 4$

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

$y = x^{2} - 4 x + \frac{x}{6} + \frac{1}{2 (3)} \cdot - 4$

Factor $2$ out of $- 4$ .

$y = x^{2} - 4 x + \frac{x}{6} + \frac{1}{2 \cdot 3} \cdot (2 \cdot - 2)$

Cancel the common factor.

$y = x^{2} - 4 x + \frac{x}{6} + \frac{1}{2 \cdot 3} \cdot (2 \cdot - 2)$

Rewrite the expression.

$y = x^{2} - 4 x + \frac{x}{6} + \frac{1}{3} \cdot - 2$

Combine $\frac{1}{3}$ and $- 2$ .

$y = x^{2} - 4 x + \frac{x}{6} + \frac{- 2}{3}$

Move the negative in front of the fraction.

$y = x^{2} - 4 x + \frac{x}{6} - \frac{2}{3}$

To write $- 4 x$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

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

Combine $- 4 x$ and $\frac{6}{6}$ .

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

Combine the numerators over the common denominator.

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

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

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

Combine $x^{2}$ and $\frac{6}{6}$ .

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

Combine the numerators over the common denominator.

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

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

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

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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

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

Multiply $3$ by $2$ .

$y = \frac{x^{2} \cdot 6 - 4 x \cdot 6 + x}{6} - \frac{2 \cdot 2}{6}$

Combine the numerators over the common denominator.

$y = \frac{x^{2} \cdot 6 - 4 x \cdot 6 + x - 2 \cdot 2}{6}$

Simplify the numerator.

Move $6$ to the left of $x^{2}$ .

$y = \frac{6 \cdot x^{2} - 4 x \cdot 6 + x - 2 \cdot 2}{6}$

Multiply $6$ by $- 4$ .

$y = \frac{6 x^{2} - 24 x + x - 2 \cdot 2}{6}$

Multiply $- 2$ by $2$ .

$y = \frac{6 x^{2} - 24 x + x - 4}{6}$

Add $- 24 x$ and $x$ .

$y = \frac{6 x^{2} - 23 x - 4}{6}$

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 $a \cdot c = 6 \cdot - 4 = - 24$ and whose sum is $b = - 23$ .

Factor $- 23$ out of $- 23 x$ .

$y = \frac{6 x^{2} - 23 x - 4}{6}$

Rewrite $- 23$ as $1$ plus $- 24$

$y = \frac{6 x^{2} + (1 - 24) x - 4}{6}$

Apply the distributive property.

$y = \frac{6 x^{2} + 1 x - 24 x - 4}{6}$

Multiply $x$ by $1$ .

$y = \frac{6 x^{2} + x - 24 x - 4}{6}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$y = \frac{(6 x^{2} + x) - 24 x - 4}{6}$

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

$y = \frac{x (6 x + 1) - 4 (6 x + 1)}{6}$

Factor the polynomial by factoring out the greatest common factor, $6 x + 1$ .

$y = \frac{(6 x + 1) (x - 4)}{6}$

Expand $(6 x + 1) (x - 4)$ using the FOIL Method.

Apply the distributive property.

$y = \frac{6 x (x - 4) + 1 (x - 4)}{6}$

Apply the distributive property.

$y = \frac{6 x \cdot x + 6 x \cdot - 4 + 1 (x - 4)}{6}$

Apply the distributive property.

$y = \frac{6 x \cdot x + 6 x \cdot - 4 + 1 x + 1 \cdot - 4}{6}$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$y = \frac{6 (x \cdot x) + 6 x \cdot - 4 + 1 x + 1 \cdot - 4}{6}$

Multiply $x$ by $x$ .

$y = \frac{6 x^{2} + 6 x \cdot - 4 + 1 x + 1 \cdot - 4}{6}$

Multiply $- 4$ by $6$ .

$y = \frac{6 x^{2} - 24 x + 1 x + 1 \cdot - 4}{6}$

Multiply $x$ by $1$ .

$y = \frac{6 x^{2} - 24 x + x + 1 \cdot - 4}{6}$

Multiply $- 4$ by $1$ .

$y = \frac{6 x^{2} - 24 x + x - 4}{6}$

Add $- 24 x$ and $x$ .

$y = \frac{6 x^{2} - 23 x - 4}{6}$

Split the fraction $\frac{6 x^{2} - 23 x - 4}{6}$ into two fractions.

$y = \frac{6 x^{2} - 23 x}{6} + \frac{- 4}{6}$

Split the fraction $\frac{6 x^{2} - 23 x}{6}$ into two fractions.

$y = \frac{6 x^{2}}{6} + \frac{- 23 x}{6} + \frac{- 4}{6}$

Cancel the common factor of $6$ .

Cancel the common factor.

$y = \frac{6 x^{2}}{6} + \frac{- 23 x}{6} + \frac{- 4}{6}$

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

$y = x^{2} + \frac{- 23 x}{6} + \frac{- 4}{6}$

Move the negative in front of the fraction.

$y = x^{2} - \frac{23 x}{6} + \frac{- 4}{6}$

Cancel the common factor of $- 4$ and $6$ .

Factor $2$ out of $- 4$ .

$y = x^{2} - \frac{23 x}{6} + \frac{2 (- 2)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$y = x^{2} - \frac{23 x}{6} + \frac{2 \cdot - 2}{2 \cdot 3}$

Cancel the common factor.

$y = x^{2} - \frac{23 x}{6} + \frac{2 \cdot - 2}{2 \cdot 3}$

Rewrite the expression.

$y = x^{2} - \frac{23 x}{6} + \frac{- 2}{3}$

Move the negative in front of the fraction.

$y = x^{2} - \frac{23 x}{6} - \frac{2}{3}$

Step 6

Find the Equation Given the Roots -1/6 , 4

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