Factoring Trinomials (ax² + bx + c)

Factoring ax squared plus bx plus c. In MathBound, the free Algebra I adventure game, Clamshell Clem asks these problems, and the only way past is to solve them.

Part of Polynomials and Factoring: chapter 8 of the standard course.

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10 questions, no sign-up, nothing saved. Climbs from the basics to harder questions.

What you will practice

Trial and Error Method. Factor a trinomial with a prime leading coefficient (limited factor combinations).
ac' Method / Grouping. Factor by finding numbers that multiply to 'ac' and add to 'b' then using factor by grouping.
Two Variables. Factor a trinomial of the form ax^2 + bxy + cy^2 with a leading coefficient > 1.
Factor GCF First. Factor out an integer GCF before factoring the resulting ax^2 + bx + c trinomial.

Worked examples

Made by the game’s own question engine. Try each one, then open its steps.

Example 1 · Trial and Error Method

What

goes

in

the

box?

5 x^{2} - 33 x + 18 = (5 x - 3) □

$(x + 6)$
$(x - 5)$
$(x - 7)$
$(x - 6)$

Show the worked steps

$a \cdot c = 90,$ $and$ $two$ $numbers$ $multiplying$ $to$ $that$ $and$ $adding$ $to$ $- 33$ $split$ $the$ $middle$ $term.$
$Grouping$ $then$ $gives$ $(5 x - 3) (x - 6) .$
$Taking$ $out$ $(5 x - 3)$ $leaves$ $(x - 6) .$

Answer: $(x - 6)$

Example 2 · ac' Method / Grouping

What

goes

in

the

box?

3 x^{2} - 22 x + 35 = (x - 5) □

$(3 x - 6)$
$(3 x - 7)$
$(3 x + 7)$
$(3 x - 8)$

Show the worked steps

$a \cdot c = 105,$ $and$ $two$ $numbers$ $multiplying$ $to$ $that$ $and$ $adding$ $to$ $- 22$ $split$ $the$ $middle$ $term.$
$Grouping$ $then$ $gives$ $(x - 5) (3 x - 7) .$
$Taking$ $out$ $(x - 5)$ $leaves$ $(3 x - 7) .$

Answer: $(3 x - 7)$

Example 3 · Two Variables

Factor:

5 x^{2} - 22 x y + 8 y^{2}

$(x - 4 y) (5 x - 2 y)$
$(x - 2 y) (5 x - 4 y)$
$(x - 3 y) (5 x - 3 y)$
$(x + 4 y) (5 x + 2 y)$

Show the worked steps

$Treat$ $y$ $as$ $part$ $of$ $the$ $constant$ $term.$
$(x - 4 y) (5 x - 2 y)$ $multiplies$ $back$ $to$ $the$ $trinomial.$

Answer: $(x - 4 y) (5 x - 2 y)$

Where it is in your textbook

Textbook	Lessons
Standard course	8.6
Big Ideas Math Algebra 1	7.6
Big Ideas Math Algebra 1 (2022)	7.6
Carnegie Learning Algebra I	M5 2.4
CPM Core Connections Algebra	8.1.3, 8.1.4
CPS Skyline Algebra	7.8H
enVision Algebra 1	7-6
enVision Algebra 1 (2024)	7-6
Eureka Math Algebra I (EngageNY)	4.3-4.4
Glencoe Algebra 1	8-7
HMH Into Algebra 1	17.3
Holt McDougal Larson Algebra 1	8.6
Illustrative Mathematics Algebra 1	7.9-7.10
Illustrative Mathematics Algebra 1 (v.360)	8.9-8.10
Math Nation Algebra 1 (Florida B.E.S.T.)	7.4, 7.5, 7.6
Open Up NC Math 1 (MVP)	7.5
OpenStax Elementary Algebra 2e	7.3
Pearson Algebra 1 Common Core	8-6
Reveal Algebra 1	10-6
SpringBoard Algebra 1	27

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