Find the Equation Given the Roots 3 , 7 , -0.5

$3$ , $7$ , $- 0.5$

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 $x = 7$ was found by solving for $x$ when $x - (7) = y$ and $y = 0$ .

The factor is $x - 7$

Step 4

The root at $x = - 0.5$ was found by solving for $x$ when $x - (- 0.5) = y$ and $y = 0$ .

The factor is $x + 0.5$

Step 5

Combine all the factors into a single equation.

$y = (x - 3) (x - 7) (x + 0.5)$

Step 6

Multiply all the factors to simplify the equation $y = (x - 3) (x - 7) (x + 0.5)$ .

Expand $(x - 3) (x - 7)$ using the FOIL Method.

Apply the distributive property.

$y = (x (x - 7) - 3 (x - 7)) (x + 0.5)$

Apply the distributive property.

$y = (x \cdot x + x \cdot - 7 - 3 (x - 7)) (x + 0.5)$

Apply the distributive property.

$y = (x \cdot x + x \cdot - 7 - 3 x - 3 \cdot - 7) (x + 0.5)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = (x^{2} + x \cdot - 7 - 3 x - 3 \cdot - 7) (x + 0.5)$

Move $- 7$ to the left of $x$ .

$y = (x^{2} - 7 \cdot x - 3 x - 3 \cdot - 7) (x + 0.5)$

Multiply $- 3$ by $- 7$ .

$y = (x^{2} - 7 x - 3 x + 21) (x + 0.5)$

Subtract $3 x$ from $- 7 x$ .

$y = (x^{2} - 10 x + 21) (x + 0.5)$

Expand $(x^{2} - 10 x + 21) (x + 0.5)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} \cdot 0.5 - 10 x \cdot x - 10 x \cdot 0.5 + 21 x + 21 \cdot 0.5$

Simplify terms.

Simplify each term.

Multiply $x^{2}$ by $x$ by adding the exponents.

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

$y = x^{2} x + x^{2} \cdot 0.5 - 10 x \cdot x - 10 x \cdot 0.5 + 21 x + 21 \cdot 0.5$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = x^{2 + 1} + x^{2} \cdot 0.5 - 10 x \cdot x - 10 x \cdot 0.5 + 21 x + 21 \cdot 0.5$

Add $2$ and $1$ .

$y = x^{3} + x^{2} \cdot 0.5 - 10 x \cdot x - 10 x \cdot 0.5 + 21 x + 21 \cdot 0.5$

Move $0.5$ to the left of $x^{2}$ .

$y = x^{3} + 0.5 \cdot x^{2} - 10 x \cdot x - 10 x \cdot 0.5 + 21 x + 21 \cdot 0.5$

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

Move $x$ .

$y = x^{3} + 0.5 x^{2} - 10 (x \cdot x) - 10 x \cdot 0.5 + 21 x + 21 \cdot 0.5$

Multiply $x$ by $x$ .

$y = x^{3} + 0.5 x^{2} - 10 x^{2} - 10 x \cdot 0.5 + 21 x + 21 \cdot 0.5$

Multiply $0.5$ by $- 10$ .

$y = x^{3} + 0.5 x^{2} - 10 x^{2} - 5 x + 21 x + 21 \cdot 0.5$

Multiply $21$ by $0.5$ .

$y = x^{3} + 0.5 x^{2} - 10 x^{2} - 5 x + 21 x + 10.5$

Simplify by adding terms.

Subtract $10 x^{2}$ from $0.5 x^{2}$ .

$y = x^{3} - 9.5 x^{2} - 5 x + 21 x + 10.5$

Add $- 5 x$ and $21 x$ .

$y = x^{3} - 9.5 x^{2} + 16 x + 10.5$

Step 7

Find the Equation Given the Roots 3 , 7 , -0.5

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