Precalculus Examples

$7$ , $8$

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

$y = 0$ at the roots

The root at $x = 7$ was found by solving for $x$ when $(x - (7)) = y$ and $y = 0$ ( $y = 0$ because the root is the x-intercept).

The factor is $x - 7$

The root at $x = 8$ was found by solving for $x$ when $(x - (8)) = y$ and $y = 0$ ( $y = 0$ because the root is the x-intercept).

The factor is $x - 8$

Combine all the factors into a single equation.

$y = (x - 7) (x - 8)$

Multiply all the factors to simplify the equation $y = (x - 7) (x - 8)$ .

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Expand $(x - 7) (x - 8)$ using the FOIL Method.

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Apply the distributive property.

$y = x (x - 8) - 7 (x - 8)$

Apply the distributive property.

$y = x x + x \cdot - 8 - 7 (x - 8)$

Apply the distributive property.

$y = x x + x \cdot - 8 + (- 7 x - 7 \cdot - 8)$

Remove parentheses.

$y = x x + x \cdot - 8 - 7 x - 7 \cdot - 8$

Simplify and combine like terms.

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Simplify each term.

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

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Combine $x$ and $x$

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

$y = x x + x \cdot - 8 - 7 x - 7 \cdot - 8$

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

$y = x x + x \cdot - 8 - 7 x - 7 \cdot - 8$

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

$y = x^{1 + 1} + x \cdot - 8 - 7 x - 7 \cdot - 8$

Add $1$ and $1$ .

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

Move $- 8$ to the left of the expression $x \cdot - 8$ .

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

Multiply $- 8$ by $x$ .

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

Multiply $- 7$ by $- 8$ .

$y = x^{2} - 8 x - 7 x + 56$

Subtract $7 x$ from $- 8 x$ .

$y = x^{2} - 15 x + 56$

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