Quadratic equations have the form ax² + bx + c = 0 and can have 0, 1, or 2 real solutions. The SAT tests factoring, the quadratic formula, and interpreting the discriminant.
Key Rules
Factor when possible: find two numbers that multiply to ac and add to b.
Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a — use when factoring is difficult.
Discriminant (b² - 4ac): positive → 2 solutions; zero → 1 solution; negative → no real solutions.
Completing the square: rewrite ax² + bx as a(x + b/2a)² - (b/2a)².
Vieta's formulas: sum of roots = -b/a; product of roots = c/a.
Common Mistakes to Avoid
Not setting the equation equal to zero before factoring.
Sign errors in the quadratic formula, especially with the ± and the -b.
Forgetting that (x-3)(x+2)=0 means x=3 OR x=-2, not x=-3 or x=2.
Squaring both sides to solve radical equations without checking for extraneous solutions.
SAT Strategy Tips
→Always move everything to one side first: ax² + bx + c = 0.
→If the question gives the roots, work backward: if roots are r and s, the equation is (x-r)(x-s)=0.
→For "no real solutions" problems: discriminant < 0, so b² - 4ac < 0.
→The SAT often uses the vertex form f(x) = a(x-h)² + k — the vertex is (h, k).
Example Question
The equation x² - 5x + 6 = 0 has solutions x = p and x = q, where p > q. What is the value of p - q?
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