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## Factor quadratics with positive coefficients

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Suppose your height above sea level in feet when traveling over a hill can be represented by the expression , where is the horizontal distance traveled. If you wanted to factor this expression, could you do it? What would be the steps that you would follow? After completing this Concept, you'll know what to do in order to factor an expression such as this one.

### Guidance

In this Concept, we will learn how to factor quadratic polynomials for different values of , and . In the last Concept, we factored common monomials, so you already know how to factor quadratic polynomials where .

Factoring Quadratic Expressions in Standard Form

Quadratic polynomials are polynomials of degree 2. The standard form of a quadratic polynomial is , where , and are real numbers.

#### Example A

Factor .

Solution: We are looking for an answer that is the product of the two binomials in parentheses: .

To fill in the blanks, we want two numbers and that multiply to 6 and add to 5. A good strategy is to list the possible ways we can multiply two numbers to give us 6 and then see which of these pairs of numbers add to 5. The number six can be written as the product of:

We can check to see if this is correct by multiplying .

is multiplied by and .

2 is multiplied by and .

Combine the like terms: .

#### Example B

Factor .

Solution: We are looking for an answer that is the product of the two binomials in parentheses: .

The number 8 can be written as the product of the following numbers.

and Notice that these are two different choices.

And

and This is the correct choice.

#### Example C

Factor .

Solution: We are looking for an answer that is the product of the two binomials in parentheses: .

In this case, we must take the negative sign into account. The number –15 can be written as the product of the following numbers.

and Notice that these are two different choices.

And also,

and Notice that these are two different choices.

and This is the correct choice.

and

To Summarize:

A quadratic of the form factors as a product of two binomials: .

• If and are positive then both and are positive.
• Example: factors as .
• If is negative and is positive then both and are negative.
• Example: factors as .
• If is negative then either is positive and is negative or vice-versa.
• Example: factors as .
• Example: factors as .
• If , factor a common factor of –1 from each term in the trinomial and then factor as usual. The answer will have the form .
• Example: factors as .

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### Guided Practice

Factor .

Solution: First factor the common factor of –1 from each term in the trinomial. Factoring –1 changes the signs of each term in the expression.

We are looking for an answer that is the product of the two binomials in parentheses: .

Now our job is to factor .

The number –6 can be written as the product of the following numbers.

### Explore More

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Factoring Quadratic Equations (16:30)

Mixed Review

1. Evaluate when .
2. Simplify .
3. Graph the following on a number line: .
4. What is the multiplicative inverse of ?

#### Quick Quiz

1. Name the following polynomial. State its degree and leading coefficient: .
2. Simplify .
3. A rectangular solid has dimensions by by . Find its volume.
4. Simplify .
5. Find the solutions to .
6. Multiply .

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 9.8.

### Vocabulary Language: English Spanish

A quadratic polynomial is a polynomial of the 2nd degree, in other words, a polynomial with an $x^2$ term.

The standard form of a quadratic polynomial is $ax^2+bx+c$, where $a,\ b$, and$c$ are real numbers.
Greatest Common Factor

Greatest Common Factor

The greatest common factor of two numbers is the greatest number that both of the original numbers can be divided by evenly.
linear factors

linear factors

Linear factors are expressions of the form $ax+b$ where $a$ and $b$ are real numbers.
Polynomial

Polynomial

A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.

The quadratic formula states that for any quadratic equation in the form $ax^2+bx+c=0$, $x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$.
Trinomial

Trinomial

A trinomial is a mathematical expression with three terms.