Activity
This activity is an extension of the expectations in the TEKS.
Here is a clever way to think about quadratic expressions that can make it easier to find their factored form.
Factor .
Step 1 - Notice the square root of the first term, , is a factor of the second term.
Step 2 - Rewrite in terms of . The third term is unchanged.
Step 3 - Substitute for the .
Step 4 - Factor the new trinomial.
Step 5 - Replace with to find the final factored form.
Using this substitution method creates a simpler trinomial that we may factor more easily.
Use the distributive property, or FOIL, to expand and simplify . Write the resulting expression in standard form. Be prepared to explain if it is equivalent to .
Enter the simplified expression and your explanation.
Compare your answer:
. Yes, the expressions are equivalent.
Study the substitution method and make sense of what was done in each step. Why might the simplified trinomial using be simpler to factor?
Compare your answer:
The trinomial using has a squared term with a coefficient of 1. Its factors can be found more easily.
In questions 3 and 4, try this -substitution method to write each of these expressions in factored form.
Enter the expression in factored form.
Compare your answer:
The square root of the first term, , is a factor of the second term, . The expression can be rewritten as . Substitute for the : . This new trinomial can be factored into . Replace with to find the final factored form: . So, .
Enter the expression in factored form.
Compare your answer:
The square root of the first term, , is a factor of the second term, . The expression can be rewritten as . Substitute for the : . This new trinomial can be factored into . Replace with to find the final factored form: . So, .
You have probably noticed that the coefficient of the squared term in all of the previous examples is a perfect square. What if that coefficient is not a perfect square?
Here is an example of an expression whose squared term has a coefficient that is not a squared term.
We see that multiplying the leading coefficient by 5 would make the first term into a perfect square, . But, we cannot introduce a new factor that will impact the value of the expression.
We need to find a way to balance the impact of multiplying everything by 5.
Knowing this, we multiply the entire trinomial by , since is the same as multiplying by 1.
Step 1 - Multiply by .
Step 2 - Distribute the 5 into the trinomial.
Step 3 - Rewrite in terms of the square root of the first term, .
Step 4 - Substitute for .
Step 5 - Factor the trinomial.
Step 6 - Replace with to find the final factored form.
Step 7 - Factor 5 out of one of the factors.
Step 8 -
Use the distributive property, or FOIL, to expand and simplify . Write the resulting expression in standard form. Be prepared to explain if it is equivalent to .
Enter the expression in standard form.
Compare your answer:
. Yes, the expressions are equivalent.
Applying the same strategy from the example to a trinomial with a leading term of , what is the multiplication expression equivalent to 1 you would use to rewrite the trinomial? Be prepared to show your reasoning.
Compare your answer:
; multiplying the leading term of the trinomial by 6 would result in a leading term of , which is a perfect square.
For questions 7 and 8, try the -method to write each of these expressions in factored form.
Enter the expression in factored form.
Compare your answer:
Multiply the expression by .
Distributing the 3 into the trinomial gives .
Rewriting the expression in terms of the square root of the first term, , gives .
Substituting for makes the expression , which can be factored into .
can now be replaced with and the expression becomes .
Factor 3 out of the first factor, because . So, .
Enter the expression in factored form.
Compare your answer:
Multiply the expression by .
Distributing the 10 into the trinomial gives . Rewriting the expression in terms of the square root of the first term, , gives . Substituting for makes the expression , which can be factored into . can now be replaced with and the expression becomes . Factor 10 out of the first factor, because . So, .
Self Check
Additional Resources
Finding the Factors of Quadratic Expressions in Standard Form
Let's look at a shortcut that can help to factor a complicated quadratic expression. This method will be useful only when the coefficient of the first term is a perfect square, such as 4 or 9.
Example 1
Step 1 - Rewrite the first two terms with as the common factor.
Step 2 - Substitute for .
Step 3 - Factor the simplified expression.
Step 4 - Replace with .
Here is an example of an expression whose squared term has a coefficient that is not a squared term. A different method can still be applied.
Example 2
Step 1 - Multiply by to make a coefficient with a squared term without impacting the overall value of the expression.
Step 2 - Distribute the 5.
Step 3 - Rewrite the first two terms with as the common factor.
Step 4 - Substitute for .
Step 5 - Factor the simplified expression.
Step 6 - Replace with .
Step 7 - Factor 5 out of one of the expressions.
Step 8 -
Step 9 - Use the distributive property, or FOIL, to expand . Is it equivalent to ?
They are equivalent.
Try it
Try It: Finding the Factors of Quadratic Expressions in Standard Form
Find the factored form of .
Here is how to use the shortcuts you just learned to find the factored form of a quadratic equation:
Step 1 - Multiply by to make a coefficient with a squared term without impacting the overall value of the expression.
Step 2 - Distribute the 3.
Step 3 - Rewrite the first two terms with as the common factor.
Step 4 - Substitute for .
Step 5 - Factor the simplified expression.
Step 6 - Replace with .
Step 7 - Factor 3 out of one of the expressions.
Step 8 -
The factored form is .