Precalculus Examples

\frac{1}{2}

$\frac{1}{2}$ ,

3

$3$

Step 1

Roots are the points where the graph intercepts with the x-axis

(y = 0)

$(y = 0)$ .

y = 0

$y = 0$ at the roots

Step 2

The root at

x = \frac{1}{2}

$x = \frac{1}{2}$ was found by solving for

x

$x$ when

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

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

y = 0

$y = 0$ .

The factor is

x - \frac{1}{2}

$x - \frac{1}{2}$

Step 3

The root at

x = 3

$x = 3$ was found by solving for

x

$x$ when

x - (3) = y

$x - (3) = y$ and

y = 0

$y = 0$ .

The factor is

x - 3

$x - 3$

Step 4

Combine all the factors into a single equation.

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

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

Step 5

Multiply all the factors to simplify the equation

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

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

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Step 5.1

Expand

(x - \frac{1}{2}) (x - 3)

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

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

Step 5.1.2

Apply the distributive property.

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

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

Step 5.1.3

Apply the distributive property.

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

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

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

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

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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