Precalculus Examples Rational Expressions and Equations

Find the Equation Given the Roots

$\frac{1}{2}$ , $3$

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 $x = \frac{1}{2}$ was found by solving for $x$ when $x - (\frac{1}{2}) = y$ and $y = 0$ .

The factor is $x - \frac{1}{2}$

Step 3

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

The factor is $x - 3$

Step 4

Combine all the factors into a single equation.

$y = (x - \frac{1}{2}) (x - 3)$

Step 5

Multiply all the factors to simplify the equation $y = (x - \frac{1}{2}) (x - 3)$ .

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Step 5.1

Expand $(x - \frac{1}{2}) (x - 3)$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

$y = x (x - 3) - \frac{1}{2} \cdot (x - 3)$

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $x$ by $x$ .

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

Step 5.2.1.2

Move $- 3$ to the left of $x$ .

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

Step 5.2.1.3

Combine $x$ and $\frac{1}{2}$ .

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

Step 5.2.1.4

Multiply $- \frac{1}{2} \cdot - 3$ .

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Step 5.2.1.4.1

Multiply $- 3$ by $- 1$ .

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

Step 5.2.1.4.2

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

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.2.4

Combine the numerators over the common denominator.

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

Step 5.2.5

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

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

Step 5.2.6

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

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

Step 5.2.7

Combine the numerators over the common denominator.

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

Step 5.2.8

Combine the numerators over the common denominator.

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

Step 5.3

Simplify the numerator.

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Step 5.3.1

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

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

Step 5.3.2

Multiply $2$ by $- 3$ .

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

Step 5.3.3

Subtract $x$ from $- 6 x$ .

$y = \frac{2 x^{2} - 7 x + 3}{2}$

Step 5.3.4

Factor by grouping.

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Step 5.3.4.1

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 $a \cdot c = 2 \cdot 3 = 6$ and whose sum is $b = - 7$ .

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Step 5.3.4.1.1

Factor $- 7$ out of $- 7 x$ .

$y = \frac{2 x^{2} - 7 x + 3}{2}$

Step 5.3.4.1.2

Rewrite $- 7$ as $- 1$ plus $- 6$

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

Step 5.3.4.1.3

Apply the distributive property.

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

Step 5.3.4.2

Factor out the greatest common factor from each group.

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Step 5.3.4.2.1

Group the first two terms and the last two terms.

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

Step 5.3.4.2.2

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

$y = \frac{x (2 x - 1) - 3 (2 x - 1)}{2}$

Step 5.3.4.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$y = \frac{(2 x - 1) (x - 3)}{2}$

Step 5.4

Expand $(2 x - 1) (x - 3)$ using the FOIL Method.

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Step 5.4.1

Apply the distributive property.

$y = \frac{2 x (x - 3) - 1 (x - 3)}{2}$

Step 5.4.2

Apply the distributive property.

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

Step 5.4.3

Apply the distributive property.

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

Step 5.5

Simplify and combine like terms.

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Step 5.5.1

Simplify each term.

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Step 5.5.1.1

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

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Step 5.5.1.1.1

Move $x$ .

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

Step 5.5.1.1.2

Multiply $x$ by $x$ .

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

Step 5.5.1.2

Multiply $- 3$ by $2$ .

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

Step 5.5.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 5.5.1.4

Multiply $- 1$ by $- 3$ .

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

Step 5.5.2

Subtract $x$ from $- 6 x$ .

$y = \frac{2 x^{2} - 7 x + 3}{2}$

Step 5.6

Split the fraction $\frac{2 x^{2} - 7 x + 3}{2}$ into two fractions.

$y = \frac{2 x^{2} - 7 x}{2} + \frac{3}{2}$

Step 5.7

Split the fraction $\frac{2 x^{2} - 7 x}{2}$ into two fractions.

$y = \frac{2 x^{2}}{2} + \frac{- 7 x}{2} + \frac{3}{2}$

Step 5.8

Cancel the common factor of $2$ .

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Step 5.8.1

Cancel the common factor.

$y = \frac{2 x^{2}}{2} + \frac{- 7 x}{2} + \frac{3}{2}$

Step 5.8.2

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

$y = x^{2} + \frac{- 7 x}{2} + \frac{3}{2}$

Step 5.9

Move the negative in front of the fraction.

$y = x^{2} - \frac{7 x}{2} + \frac{3}{2}$

Step 6

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Find the Equation Given the Roots

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