📝 Quadratic Equations

Part 1 of 7 — Quadratic Form & Factoring

Standard form: ax² + bx + c = 0.

Factoring: find two numbers that multiply to ac and add to b.

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Key Insight: Zero product property: if (x - r)(x - s) = 0, then x = r or x = s.

SAT Tip: Special patterns: difference of squares a² - b² = (a+b)(a-b), perfect square trinomials.

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Completing the Square

Part 2 of 7 — Completing the Square

Completing the square: x² + bx + (b/2)² = (x + b/2)².

Move the constant to the other side first.

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Key Insight: Add (b/2)² to both sides.

SAT Tip: Vertex form: a(x - h)² + k where (h, k) is the vertex.

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The Quadratic Formula

Part 3 of 7 — The Quadratic Formula

x = (-b ± √(b² - 4ac)) / (2a).

Discriminant b² - 4ac determines the number of real solutions.

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Key Insight: b² - 4ac > 0: two real solutions; = 0: one (repeated); < 0: no real solutions.

SAT Tip: SAT Tip: the discriminant alone can answer "how many solutions?" questions.

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Graphing Parabolas

Part 4 of 7 — Graphing Parabolas

Parabola opens up if a > 0, down if a < 0.

Vertex: x = -b/(2a), then find y.

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Key Insight: Axis of symmetry: x = -b/(2a).

SAT Tip: x-intercepts (roots): set y = 0 and solve; y-intercept: set x = 0.

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Quadratic Word Problems

Part 5 of 7 — Quadratic Word Problems

Projectile motion: h(t) = -16t² + v₀t + h₀ (feet) or h(t) = -4.9t² + v₀t + h₀ (meters).

Maximum height: find the vertex.

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Key Insight: When does it hit the ground? Set h(t) = 0 and solve.

SAT Tip: Area problems: set up quadratic from length/width relationships.

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Problem-Solving Workshop

Part 6 of 7 — Problem-Solving Workshop

Projectile motion: h(t) = -16t² + v₀t + h₀ (feet) or h(t) = -4.9t² + v₀t + h₀ (meters).

Maximum height: find the vertex.

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Key Insight: When does it hit the ground? Set h(t) = 0 and solve.

SAT Tip: Area problems: set up quadratic from length/width relationships.

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Review & Applications

Part 7 of 7 — Review & Applications

Projectile motion: h(t) = -16t² + v₀t + h₀ (feet) or h(t) = -4.9t² + v₀t + h₀ (meters).

Maximum height: find the vertex.

Check Your Understanding 🎯

Key Insight: When does it hit the ground? Set h(t) = 0 and solve.

SAT Tip: Area problems: set up quadratic from length/width relationships.

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Match the Concepts 🔍