Factoring Quadratic Expressions
Factoring rewrites a quadratic as a product of two binomials — the key step behind solving many SAT quadratic questions.
Reading the concept isn't enough. The score comes from practicing the real Bluebook-style format and finding the exact mistakes you keep making.
What the SAT tests here
For $x^2 + bx + c$, find two numbers that multiply to $c$ and add to $b$. When there's a leading coefficient, account for it too — don't factor the constant alone.
How to solve one, step by step
Example: factor $x^2 + 5x + 6$.
- Find two numbers that multiply to $6$ and add to $5$: that's $2$ and $3$.
- So $x^2 + 5x + 6 = (x + 2)(x + 3)$.
The mistakes that cost points
- Ignoring the leading coefficient. When the $x^2$ term has a coefficient, factoring the constant alone gives the wrong binomials.
- Sign errors in the factors. Match the signs so the product and sum both come out right.
Practice questions
Try these the way you would on test day, then open the solution to check your method.
Easy
In a class, $\frac{k}{30} = \frac{8}{24}$. What is the value of $k$?
- A$k = 8$
- B$k = 10$
- C$k = 240$
- D$k = 16$
Show solution
Answer: B, $k = 10$. $\frac{8}{24} = \frac{1}{3}$. So $\frac{k}{30} = \frac{1}{3} \Rightarrow k = 10$.
Medium
A map uses a scale of $2$ cm $= 50$ km. If two cities are $k$ cm apart on the map and 175 km apart in reality, what is the value of $k$?
- A$k = 2$
- B$k = 350$
- C$k = 7$
- D$k = 9$
Show solution
Answer: C, $k = 7$. $\frac{2}{50} = \frac{k}{175} \Rightarrow 2 \times 175 = 50k \Rightarrow 350 = 50k \Rightarrow k = 7$.
Hard
A recipe for 4 servings uses $\frac{3}{4}$ cup of oil. To make $k$ servings using $3$ cups of oil, what is the value of $k$?
- A$k = 7$
- B$k = 36$
- C$k = 16$
- D$k = 4$
Show solution
Answer: C, $k = 16$. $\frac{3/4}{4} = \frac{3}{k} \Rightarrow \frac{3}{4} \cdot k = 4 \cdot 3 = 12 \Rightarrow k = 16$.
Find your exact gaps
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