Find the Equation Given the Roots 3 , -5/2

$3$ , $- \frac{5}{2}$

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 $x = - \frac{5}{2}$ was found by solving for $x$ when $x - (- \frac{5}{2}) = y$ and $y = 0$ .

The factor is $x + \frac{5}{2}$

Step 4

Combine all the factors into a single equation.

$y = (x - 3) (x + \frac{5}{2})$

Step 5

Multiply all the factors to simplify the equation $y = (x - 3) (x + \frac{5}{2})$ .

Expand $(x - 3) (x + \frac{5}{2})$ using the FOIL Method.

Apply the distributive property.

$y = x (x + \frac{5}{2}) - 3 (x + \frac{5}{2})$

Apply the distributive property.

$y = x \cdot x + x (\frac{5}{2}) - 3 (x + \frac{5}{2})$

Apply the distributive property.

$y = x \cdot x + x (\frac{5}{2}) - 3 x - 3 (\frac{5}{2})$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = x^{2} + x (\frac{5}{2}) - 3 x - 3 (\frac{5}{2})$

Combine $x$ and $\frac{5}{2}$ .

$y = x^{2} + \frac{x \cdot 5}{2} - 3 x - 3 (\frac{5}{2})$

Move $5$ to the left of $x$ .

$y = x^{2} + \frac{5 \cdot x}{2} - 3 x - 3 (\frac{5}{2})$

Multiply $- 3 (\frac{5}{2})$ .

Combine $- 3$ and $\frac{5}{2}$ .

$y = x^{2} + \frac{5 x}{2} - 3 x + \frac{- 3 \cdot 5}{2}$

Multiply $- 3$ by $5$ .

$y = x^{2} + \frac{5 x}{2} - 3 x + \frac{- 15}{2}$

Move the negative in front of the fraction.

$y = x^{2} + \frac{5 x}{2} - 3 x - \frac{15}{2}$

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

$y = x^{2} + \frac{5 x}{2} - 3 x \cdot \frac{2}{2} - \frac{15}{2}$

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

$y = x^{2} + \frac{5 x}{2} + \frac{- 3 x \cdot 2}{2} - \frac{15}{2}$

Combine the numerators over the common denominator.

$y = x^{2} + \frac{5 x - 3 x \cdot 2}{2} - \frac{15}{2}$

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

$y = x^{2} \cdot \frac{2}{2} + \frac{5 x - 3 x \cdot 2}{2} - \frac{15}{2}$

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

$y = \frac{x^{2} \cdot 2}{2} + \frac{5 x - 3 x \cdot 2}{2} - \frac{15}{2}$

Combine the numerators over the common denominator.

$y = \frac{x^{2} \cdot 2 + 5 x - 3 x \cdot 2}{2} - \frac{15}{2}$

Combine the numerators over the common denominator.

$y = \frac{x^{2} \cdot 2 + 5 x - 3 x \cdot 2 - 15}{2}$

Simplify the numerator.

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

$y = \frac{2 \cdot x^{2} + 5 x - 3 x \cdot 2 - 15}{2}$

Multiply $2$ by $- 3$ .

$y = \frac{2 x^{2} + 5 x - 6 x - 15}{2}$

Subtract $6 x$ from $5 x$ .

$y = \frac{2 x^{2} - x - 15}{2}$

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 $a \cdot c = 2 \cdot - 15 = - 30$ and whose sum is $b = - 1$ .

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

$y = \frac{2 x^{2} - (x) - 15}{2}$

Rewrite $- 1$ as $5$ plus $- 6$

$y = \frac{2 x^{2} + (5 - 6) x - 15}{2}$

Apply the distributive property.

$y = \frac{2 x^{2} + 5 x - 6 x - 15}{2}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$y = \frac{(2 x^{2} + 5 x) - 6 x - 15}{2}$

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

$y = \frac{x (2 x + 5) - 3 (2 x + 5)}{2}$

Factor the polynomial by factoring out the greatest common factor, $2 x + 5$ .

$y = \frac{(2 x + 5) (x - 3)}{2}$

Expand $(2 x + 5) (x - 3)$ using the FOIL Method.

Apply the distributive property.

$y = \frac{2 x (x - 3) + 5 (x - 3)}{2}$

Apply the distributive property.

$y = \frac{2 x \cdot x + 2 x \cdot - 3 + 5 (x - 3)}{2}$

Apply the distributive property.

$y = \frac{2 x \cdot x + 2 x \cdot - 3 + 5 x + 5 \cdot - 3}{2}$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$y = \frac{2 (x \cdot x) + 2 x \cdot - 3 + 5 x + 5 \cdot - 3}{2}$

Multiply $x$ by $x$ .

$y = \frac{2 x^{2} + 2 x \cdot - 3 + 5 x + 5 \cdot - 3}{2}$

Multiply $- 3$ by $2$ .

$y = \frac{2 x^{2} - 6 x + 5 x + 5 \cdot - 3}{2}$

Multiply $5$ by $- 3$ .

$y = \frac{2 x^{2} - 6 x + 5 x - 15}{2}$

Add $- 6 x$ and $5 x$ .

$y = \frac{2 x^{2} - x - 15}{2}$

Split the fraction $\frac{2 x^{2} - x - 15}{2}$ into two fractions.

$y = \frac{2 x^{2} - x}{2} + \frac{- 15}{2}$

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

$y = \frac{2 x^{2}}{2} + \frac{- x}{2} + \frac{- 15}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$y = \frac{2 x^{2}}{2} + \frac{- x}{2} + \frac{- 15}{2}$

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

$y = x^{2} + \frac{- x}{2} + \frac{- 15}{2}$

Move the negative in front of the fraction.

$y = x^{2} - \frac{x}{2} + \frac{- 15}{2}$

Move the negative in front of the fraction.

$y = x^{2} - \frac{x}{2} - \frac{15}{2}$

Step 6

Find the Equation Given the Roots 3 , -5/2

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