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Section 3.2 Intercepts and Factors

In this lesson we review the skills we need to solve quadratic equations by factoring.

Subsection 1. Multiply binomials

Subsubsection Examples

Example 3.15.

Expand the product \(~(2x-3)(x-6)~\text{.}\)

Solution

Multiply each term of the first binomial be each term of the second binomial. This gives four multiplications, often denoted by "FOIL," which stands for First terms, Outside terms, Inside terms, and Last terms.

\begin{align*} (2x+3)(x-6) \amp = \underline{2x \cdot x}~+~\underline{2x \cdot (-6)}~+~\underline{(-3) \cdot x}~+~\underline{(-3) \cdot (-6)}\\ \amp \quad \quad \alert{\text{F}}\quad \quad \quad \quad \alert{\text{O}} \quad \quad \quad \quad \quad \alert{\text{I}} \quad \quad \quad \quad \quad \alert{\text{L}}\\ \amp = 2x^2-12x-3x+18 ~~~~~~~~\quad\blert{\text{Combine like terms.}}\\ \amp = 2x^2-15x+18 \end{align*}
Example 3.16.

Expand the product \(~-2(3x-4)(3x-5)\text{.}\)

Solution

First, multiply the binomial factors together.

\begin{equation*} (3x-4)(3x-5) = 9x^2-27x+20 \end{equation*}

Then use the distributive law to multiply the result by the monomial factor, \(-2\text{.}\)

\begin{equation*} \alert{-2}(9x^2-27x+20) = -18x^2+54x-40 \end{equation*}

Subsubsection Exercises

Expand the product \(~(2x+1)(3x-2)\text{.}\)

Answer
\(6x^2-x-2\)

Expand the product \(~(2t+5)(2t+5)\text{.}\)

Answer
\(4t^2+20t+25\)

Expand the product \(~4(a-3)(3a-5)\text{.}\)

Answer
\(12a^2-56a+60\)

Expand the product \(~-3(2b-3)(5b+1)\text{.}\)

Answer
\(-30b^2+39b+9\)

Subsection 2. Factor quadratic trinomials

To factor the trinomial \(x^2+bx+c\text{,}\) we look for two numbers \(p\) and \(q\) whose product \(pq\) is the constant term and whose sum \(p+q\) is the coefficient of the middle term.

\begin{equation*} \begin{aligned}[t] (x+p)(x+q) \amp = x^2+qx+px+pq\\ \amp = x^2 +\blert{(p+q)}x +\alert{pq} = x^2+\blert{b}x+\alert{c} \end{aligned} \end{equation*}
Sign Patterns for Quadratic Trinomials.

Assume that \(b,~c,~p\) and \(q\) are positive integers. Then

  1. \(x^2+bx+c = (x+p)(x+q)\)

    If all the coefficients of the trinomial are positive, then both \(p\) and \(q\) are positive.

  2. \(x^2-bx+c = (x-p)(x-q)\)

    If the middle term of the trinomial is negative and the other two terms are positive, then \(p\) and \(q\) are both negative.

  3. \(x^2 \pm bx-c = (x+p)(x-q)\)

    If the constant term of the trinomial is negative, then \(p\) and \(q\) have opposite signs.

Subsubsection Examples

Example 3.21.

Factor \(~t^2+7t+12~\) as a product of two binomials,

\begin{equation*} t^2+7t+12 = (t+p)(t+q) \end{equation*}
Solution

The constant term is 12, so we look for two numbers \(p\) and \(q\) whose product is 12. There are three possibilities:

\begin{equation*} \text{ 1 and 12, 2 and 6, or 3 and 4} \end{equation*}

Because the middle term is \(7t\text{,}\) we must have \(p+q=7\text{.}\) We check each possibility and find that \(p=3\) and \(q=4\text{.}\) Thus,

\begin{equation*} t^2+7t+12 = (t+3)(t+4) \end{equation*}
Example 3.22.

Factor \(~x^2-12x+20~\text{.}\)

Solution

For this example we must find two numbers \(p\) and \(q\) for which \(pq=20\) and \(p+q=-12\text{.}\) These two conditions tell us that \(p\) and \(q\) must both be negative. We start by listing all the ways to factor 20 with negative factors:

\begin{equation*} -1~~\text{and}~~-20,~~~~-2~~\text{and}~~-10,~~~~-4~~\text{and}~~-5 \end{equation*}

We check \(p+q\) for each possibility to see which one gives the correct middle term. Because \(-2+(-10) = -12\text{,}\) the factorization is

\begin{equation*} x^2-12x+20 = (x-2)(x-10) \end{equation*}
Example 3.23.

Factor \(~x^2+2x-15~\text{.}\)

Solution

This time the product \(pq\) must be negative, so \(p\) and \(q\) must have opposite signs, one positive and one negative. There are only two ways to factor 15, either 1 times 15 or 3 times 5. We just "guess" that the second factor is negative, and check \(p+q\) for each possibility:

\begin{equation*} 1-15=-14~~~~~~\text{or}~~~~~~3-5=-2 \end{equation*}

The middle term we want is \(2x\text{,}\) not \(-2x\text{,}\) so we change the signs of \(p\) and \(q\text{:}\) we use \(-3\) and \(+5\text{.}\) The correct factorization is

\begin{equation*} x^2+2x-15 = (x-3)(x+5) \end{equation*}

Subsubsection Exercises

Factor \(~x^2+8x+15\)

Answer
\((x+3)(x+5)\)

Factor \(~y^2+14y+49\)

Answer
\((y+7)(y+7)\)

Factor \(~m^2-10m+24\)

Answer
\((m-4)(m-6)\)

Factor \(~m^2-11m+24\)

Answer
\((m-3)(m-8)\)

Factor \(~t^2+8t-48\)

Answer
\((t+12)(t-4)\)

Factor \(~t^2-8t-48\)

Answer
\((t-12)(t+4)\)

Subsection 3. Write algebraic expressions

Subsubsection Examples

Example 3.30.

Ralph and Wanda together weigh 320 pounds. If Ralph weighs \(x\) pounds, how much does Wanda weigh?

Solution

We subtract Ralph's weight from the total; the remainder is Wanda's weight: \(320-x\) pounds

Example 3.31.

Delbert and Francine live 24 miles apart on Route 30. They meet at a cafe between their houses. If Delbert drove \(d\) miles, how far did Francine drive?

Solution

We subtract Delbert's distance from the total; the remainder is Francine's distance: \(24-d\) miles

Example 3.32.

Three eggs and two slices of buttered toast contain 526 calories. If one egg contains \(c\) calories, how many calories are in a slice of buttered toast?

Solution

We subtract the calories in three eggs from the total; the remainder is the number of calories in two slices of toast, so one slice has half that many calories: \(~\frac{1}{2}(526-3c)\)

Example 3.33.

The perimeter of a large rectangular playground is 124 yards. If its width is \(w\) yards, what is its length?

Solution

We subtract twice the length from the perimeter; the remainder is twice the width, so the width is half that: \(~\frac{1}{2}(124-2w) = 62-w\) yards

Subsubsection Exercises

Garth and Taylor together made $86,000 last year. If Garth made \(d\) dollars, how much did Taylor make?

Answer

\(86,000 - d\) dollars

Six coffees and four pastries cost the office manager $21. If a pastry costs \(x\) dollars, how much does a coffee cost?

Answer

\(\dfrac{1}{6}(21-4x)\) dollars

The vertex angle of an isosceles triangle is \(v\) degrees. What is the measure of each of the two base angles?

Answer

\(\dfrac{1}{2}(180-v)\) degrees

The perimeter of a rectangular swimming pool is 260 feet. If the length of the pool is \(L\) feet, what is its width?

Answer

\(\dfrac{1}{2}(260-2L)\) feet