**GMAT Focus Edition: Quadratic Equations – Complete Study Guide**

GMAT Focus Edition: Quadratic Equations – Complete Study Guide

By a 700+ GMAT Instructor

What This Is

Quadratic equations appear in ~10% of GMAT Quant questions (Problem Solving and Data Sufficiency). They test your ability to: - Factor, expand, or solve quadratics efficiently.
- Recognize hidden quadratics (e.g., (x^4 - 5x^2 + 4 = 0)).
- Apply the Zero Product Property and Quadratic Formula under time pressure.

Why it matters: Quadratics are a high-leverage topic—mastering them saves 30+ seconds per question, freeing time for harder problems. A single careless error here can cost you 10+ percentile points.

Real GMAT-Style Example:
If (x^2 - 5x + 6 = 0), which of the following could be the value of (x)? (A) -3 (B) -2 (C) 1 (D) 2 (E) 5

(Answer: D. The equation factors to ((x-2)(x-3)=0), so (x=2) or (3).)

Key Concepts & Techniques

1. Factoring (FOIL Method)
2. What it is: Expressing (ax^2 + bx + c) as ((dx + e)(fx + g)).
3. When to use: When the quadratic is factorable (check if (a=1) or if the product (a \times c) has factors that sum to (b)).

4. Example: (x^2 - 5x + 6 = (x-2)(x-3)).

5. Zero Product Property

6. What it is: If (A \times B = 0), then (A=0) or (B=0).
7. When to use: After factoring to find roots.

8. Example: ((x-2)(x-3)=0 \implies x=2) or (x=3).

9. Quadratic Formula

10. What it is: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}).
11. When to use: When factoring is too slow or impossible (e.g., (x^2 + 4x + 2 = 0)).

12. Pro Tip: Memorize the discriminant ((D = b^2 - 4ac)):

    
    
    • (D > 0): 2 real roots.
    • (D = 0): 1 real root.
    • (D < 0): No real roots (GMAT rarely tests this).

13. Completing the Square

14. What it is: Rewriting (ax^2 + bx + c) as ((x + d)^2 + e).
15. When to use: For vertex form (e.g., "What is the minimum value of (x^2 + 6x + 10)?") or when the quadratic formula is overkill.

16. Example: (x^2 + 6x + 10 = (x+3)^2 + 1) (minimum value = 1).

17. Hidden Quadratics

18. What it is: Equations like (x^4 - 5x^2 + 4 = 0) or (\frac{1}{x^2} - \frac{5}{x} + 6 = 0).
19. When to use: When you see exponents or reciprocals that can be substituted (e.g., let (y = x^2) or (y = \frac{1}{x})).

20. Example: (x^4 - 5x^2 + 4 = 0 \implies y^2 - 5y + 4 = 0) (where (y = x^2)).

21. Sum/Product of Roots

22. What it is: For (ax^2 + bx + c = 0), sum of roots = (-b/a), product = (c/a).
23. When to use: On Data Sufficiency to avoid solving (e.g., "What is the sum of the roots?").

24. Example: For (x^2 - 5x + 6 = 0), sum = 5, product = 6.

25. Special Products

26. What it is: Recognizing patterns like:
    • ((a + b)^2 = a^2 + 2ab + b^2)
    • ((a - b)^2 = a^2 - 2ab + b^2)
    • ((a + b)(a - b) = a^2 - b^2)
27. When to use: To expand or factor quickly (e.g., (x^2 - 9 = (x+3)(x-3))).

Step-by-Step Strategy

Follow this process for every quadratic equation question:

1. Simplify the Equation
2. Move all terms to one side to set the equation to 0 (e.g., (x^2 = 5x - 6 \implies x^2 - 5x + 6 = 0)).

3. Divide by common factors if (a \neq 1) (e.g., (2x^2 - 8x + 6 = 0 \implies x^2 - 4x + 3 = 0)).

4. Check for Factoring

5. If (a = 1), find two numbers that multiply to (c) and add to (b).
6. If (a \neq 1), use the AC method: Multiply (a \times c), find factors that add to (b), then split the middle term.

7. Example: (2x^2 + 7x + 3 = 0)

   
   
   • (a \times c = 6), factors = 6 and 1 (sum = 7).
   • Rewrite: (2x^2 + 6x + x + 3 = 0).
   • Factor: (2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3) = 0).

8. Apply the Zero Product Property

9. Set each factor to 0 and solve.

10. Example: ((2x + 1)(x + 3) = 0 \implies x = -1/2) or (x = -3).

11. Use the Quadratic Formula if Necessary

12. If factoring is too slow or impossible, plug into (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}).

13. Example: (x^2 + 4x + 2 = 0)

    
    
    • (a=1, b=4, c=2).
    • (x = \frac{-4 \pm \sqrt{16 - 8}}{2} = \frac{-4 \pm \sqrt{8}}{2} = -2 \pm \sqrt{2}).

14. Check for Hidden Quadratics

15. Look for substitution opportunities (e.g., (x^4, \frac{1}{x}, \sqrt{x})).

16. Example: (x - 5\sqrt{x} + 6 = 0)

    
    
    • Let (y = \sqrt{x}), then (y^2 - 5y + 6 = 0).
    • Solve: (y = 2) or (y = 3).
    • Back-substitute: (\sqrt{x} = 2 \implies x = 4) or (\sqrt{x} = 3 \implies x = 9).

17. Verify Solutions

18. Plug roots back into the original equation to check for extraneous solutions (especially for hidden quadratics).
19. Example: For (x = 4) in (x - 5\sqrt{x} + 6 = 0):
    • (4 - 5(2) + 6 = 0) ✓.

Fully Worked GMAT-Style Example

Question:
If (x^2 - 8x + k = 0) has two distinct real roots, which of the following could be the value of (k)? (A) 12 (B) 16 (C) 20 (D) 24 (E) 32

Step-by-Step Solution:

1. Simplify the Equation:

2. Already in standard form: (x^2 - 8x + k = 0).

3. Check for Distinct Real Roots:

4. For two distinct real roots, the discriminant must be positive:
   (D = b^2 - 4ac > 0).
5. Here, (a=1, b=-8, c=k).

6. So, ((-8)^2 - 4(1)(k) > 0 \implies 64 - 4k > 0 \implies 4k < 64 \implies k < 16).

7. Evaluate Answer Choices:

8. Only (A) 12 satisfies (k < 16).

Answer: A.

Common Mistakes

1. Mistake: Forgetting to set the equation to 0 before factoring.
2. Why it happens: Students rush and try to factor (x^2 = 5x - 6) directly.

3. Correct approach: Always rewrite as (x^2 - 5x + 6 = 0) first.

4. Mistake: Misapplying the Quadratic Formula (e.g., forgetting the (\pm) or the denominator).

5. Why it happens: Careless arithmetic under time pressure.

6. Correct approach: Write the formula every time and double-check signs.

7. Mistake: Ignoring extraneous solutions for hidden quadratics.

8. Why it happens: Students forget to verify roots (e.g., (\sqrt{x} = -2) is invalid).

9. Correct approach: Always plug solutions back into the original equation.

10. Mistake: Assuming all quadratics factor neatly.

11. Why it happens: Over-reliance on factoring without checking the discriminant.

12. Correct approach: If factoring takes >10 seconds, use the quadratic formula.

13. Mistake: Confusing sum/product of roots with the roots themselves.

14. Why it happens: Misapplying the formulas (-b/a) and (c/a).
15. Correct approach: Sum = (-b/a), product = (c/a) (not the roots!).

GMAT Traps & Timing

1. Trap: "Must Be True" vs. "Could Be True"
2. What it is: Questions like "Which of the following must be true?" vs. "Which could be true?".
3. How to avoid: For "must," the statement must hold for all roots. For "could," it only needs to hold for one root.

4. Example: If (x^2 - 5x + 6 = 0), then (x = 2) could be true, but (x > 0) must be true.

5. Trap: Non-Integer Roots

6. What it is: The GMAT often gives quadratics with irrational roots (e.g., (x^2 + 4x + 2 = 0)) to waste your time.

7. How to avoid: If the discriminant isn’t a perfect square, skip factoring and use the quadratic formula.

8. Trap: Data Sufficiency Over-Solving

9. What it is: Solving for roots when you only need to know if they’re real/distinct.
10. How to avoid: On DS, stop at the discriminant (e.g., "Is (D > 0)?").

Time Budget:
- Problem Solving: 1–1.5 minutes.
- Data Sufficiency: 30–45 seconds (use discriminant/sum/product shortcuts).

Quick Practice

1. Question:
   If (x^2 - 7x + 12 = 0), what is the sum of the roots?
   (A) -7 (B) -4 (C) 4 (D) 7 (E) 12

Answer: D.
Solution: Sum of roots = (-b/a = -(-7)/1 = 7).

1. Question:
   Which of the following equations has no real solutions?
   (A) (x^2 - 4x + 4 = 0) (B) (x^2 - 4x + 5 = 0) (C) (x^2 - 4x - 5 = 0)

Answer: B.
Solution: Discriminant (D = (-4)^2 - 4(1)(5) = 16 - 20 = -4 < 0) (no real roots).

Last-Minute Cram Sheet

1. Standard Form: (ax^2 + bx + c = 0).
2. Factoring Shortcut: If (a=1), find two numbers that multiply to (c) and add to (b).
3. Quadratic Formula: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}).
4. Discriminant: (D = b^2 - 4ac) → (D > 0): 2 roots, (D = 0): 1 root, (D < 0): 0 roots.
5. Sum of Roots: (-b/a).
6. Product of Roots: (c/a).
7. Hidden Quadratics: Substitute (y = x^2) or (y = \frac{1}{x}) if exponents/reciprocals appear.
8. Special Products: ((a \pm b)^2 = a^2 \pm 2ab + b^2), ((a+b)(a-b) = a^2 - b^2).
9. GMAT Trap: "Must be true" ≠ "Could be true" (check all roots vs. one root).
10. Time-Saver: On DS, stop at the discriminant—don’t solve unless necessary.

⚠️ Final Warning:
- Never assume roots are integers (GMAT loves irrational roots).
- Always check for extraneous solutions in hidden quadratics.
- Memorize the quadratic formula—it’s faster than struggling with factoring.