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Algebra basics (ASL)

Course: Algebra basics (ASL) > Unit 7

Lesson 5: Factoring quadratics 1 (ASL)
Math
Algebra basics (ASL)
Quadratics and polynomials (ASL)
Factoring quadratics 1 (ASL)
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Factoring simple quadratics review

Google Classroom
Factoring quadratics is very similar to multiplying binomials, just going the other way. For example, x^2+3x+2 factors to (x+1)(x+2) because (x+1)(x+2) multiplies to x^2+3x+2. This article reviews the basics of how to factor quadratics into the product of two binomials.

Example

Factor as the product of two binomials.
x, squared, plus, 3, x, plus, 2
Our goal is to rewrite the expression in the form:
left parenthesis, x, plus, a, right parenthesis, left parenthesis, x, plus, b, right parenthesis
Expanding left parenthesis, x, plus, a, right parenthesis, left parenthesis, x, plus, b, right parenthesis gives us a clue.
x2+3x+2=(x+a)(x+b)=x2+ax+bx+ab=x2+(a+b)x+ab\begin{aligned} x^2+\goldD{3}x+\blueD{2}&=(x+a)(x+b) \\\\ &= x^2 +ax+bx + ab \\\\ &= x^2 +\goldD{(a+b)}x + \blueD{ab} \end{aligned}
So start color #e07d10, left parenthesis, a, plus, b, right parenthesis, equals, 3, end color #e07d10 and start color #11accd, a, b, equals, 2, end color #11accd.
After playing around with different possibilities for a and b, we discover that a, equals, start color #1fab54, 1, end color #1fab54, b, equals, start color #1fab54, 2, end color #1fab54 satisfies both conditions.
Plugging these in, we get:
left parenthesis, x, plus, start color #1fab54, 1, end color #1fab54, right parenthesis, left parenthesis, x, plus, start color #1fab54, 2, end color #1fab54, right parenthesis
And we can multiply the binomials to check our solution if we'd like:
 (x+1)(x+2)= x2+2x+x+2= x2+3x+2\begin{aligned} &~(x+1)(x+2) \\\\ =&~x^2+2x+x+2 \\\\ =&~x^2+3x+2 \end{aligned}
Yep, we get our original expression back, so we know we factored correctly to get our answer:
left parenthesis, x, plus, start color #1fab54, 1, end color #1fab54, right parenthesis, left parenthesis, x, plus, start color #1fab54, 2, end color #1fab54, right parenthesis
Want to see another example? Check out this video.

Practice

Factor the quadratic expression as the product of two binomials.
x, squared, minus, x, minus, 42, equals

Want more practice? Check out this exercise.