Factoring Quadratics on the SAT
Turn Products into Solutions
Build equations from context, spot patterns fast, and practice with intent.
Why the SAT Emphasizes Factoring Quadratics
Factoring is the SAT shortcut for many quadratic equations. Instead of using a long formula, you look for two numbers that multiply and add in a specific way, then the solutions appear immediately.
This lesson walks you through the logic of factor pairs and the zero product rule. You will also see how to recognize when factoring is not efficient so you can switch to another method without wasting time.
A Simple Definition Unlocks Factoring Quadratics
When a quadratic can be written as $(x + a)(x + b) = 0$, the only way for the product to be zero is for one factor to be zero. That gives you two solutions, $x = -a$ and $x = -b$.
The challenge is finding the right pair of numbers. You need a pair that multiplies to $c$ and adds to $b$ in $x^2 + bx + c$. If no pair works, the quadratic does not factor nicely.
Work Through Factoring Quadratics Step by Step
Factor $x^2 + 5x + 6$ and solve $x^2 + 5x + 6 = 0$.
Work through a clean factoring example so you can see how the zero product rule applies.
Start with the quadratic equation
Find two numbers that multiply to $6$ and add to $5
Factor the trinomial into two binomials.
Set the first factor to zero
Set the second factor to zero
Solve the first equation to isolate the variable.
Solve the second equation to isolate the variable.
Use Desmos to Check Factoring Quadratics
Factor $x^2 + 5x + 6$ and solve $x^2 + 5x + 6 = 0$.
Desmos can solve factoring questions by graphing the quadratic and finding the $x$ intercepts. This is fast when factoring is difficult or when coefficients are large.
y = x^2 + 5x + 6
Factoring is faster when the numbers are simple. Desmos is faster when factoring is slow or when you want a quick check.
Expert move: Graph the quadratic and click the $x$-intercepts and vertex to read solutions and key features; the graph makes it clear when there are 0, 1, or 2 real roots.
Precision check: Use Desmos for decimal answers or verification, but convert to a fraction if the choices are exact and apply any context restrictions.
- Desmos features used: graphing, $x$ intercepts.
- Common mistake: reading $y$ values instead of the $x$ intercepts.
Practice Factoring Quadratics with SAT-Style Questions
Solve each quadratic by factoring when possible.
Solve .
Factor .
Solve .
If , which value of is a solution?
Key Takeaways to Remember for Factoring Quadratics
- Factor by finding two numbers that multiply to $c$ and add to $b$.
- Use the zero product rule to solve.
- Desmos is a fast check when factoring is slow.
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