Question

name: date: topic: solving quadratic equations by factoring trinomials lesson synthesis complete the sentences using the word bank below. word bank zero product property factor solutions trinomial multiply equal 1. to solve a quadratic equation like ( x^2 + 5x + 6 = 0 ), we first ____ the __ on the left side. 2. when we get a factored form like ( (x + 2)(x + 3) = 0 ), we use the __ to solve it: if two numbers __, then at least one must be zero. 3. setting each factor __ to zero gives us our ____, which are the values of ( x ) where the equation is true.

Explanation:

Question 1

To solve a quadratic equation in the form of a trinomial (like \(x^2 + 5x + 6 = 0\)), the first step is to factor the trinomial. So we need to find the appropriate words from the word bank. The first blank should be "factor" (the action) and the second blank should be "trinomial" (the expression being factored).

Step 1: Analyze the first blank

The action taken to solve \(x^2 + 5x + 6 = 0\) initially is factoring, so the first word is "factor".

Step 2: Analyze the second blank

The expression \(x^2 + 5x + 6\) is a trinomial (a polynomial with three terms), so the second word is "trinomial".

When we have a factored form like \((x + 2)(x + 3)=0\), we use the zero product property. The zero product property states that if two numbers multiply to zero, then at least one of them must be zero. So we identify the correct terms from the word bank.

Step 1: Analyze the first blank

The property used here is the "zero product property".

Step 2: Analyze the second blank

The zero product property involves the idea that if two numbers multiply to zero, so the second blank is "multiply".

After factoring and applying the zero product property, we set each factor equal to zero to find the solutions (the values of \(x\) that satisfy the equation).

Step 1: Analyze the first blank

The action is setting the factor equal to zero, so the first word is "equal".

Step 2: Analyze the second blank

The values we find by setting factors equal to zero are the "solutions" of the equation.

Answer:

1. factor; trinomial

Question 2