Quadratics

Quadratics show up in both Part I and the longer constructed-response questions. Worth knowing cold.

Click here for some tips about quadratics

Quadratic questions come in a few forms: factoring a polynomial, solving by factoring or the quadratic formula, interpreting a parabola from a graph, and working with real-world height problems. The golf ball problem is practically a Regents staple at this point.

• Your toolkit

• Standard form: ax² + bx + c = 0

  
  

• Axis of symmetry (x-coordinate of vertex): x = −b / (2a)

  
  

• Quadratic formula: x = (−b ± √(b² − 4ac)) / (2a)

  
  

• "Factor completely" = pull out the GCF first, then factor the remaining trinomial.

  
  

✓Height problems always follow the same pattern. When does it hit the ground? Set the expression equal to 0 and solve. Max height? Find t = −b / (2a), substitute back in. What does the constant represent? It's the height at t = 0.

Try it yourself!:

Warm-up problem (Level 1)

Problem: Solve by factoring: x² − 5x + 6 = 0

Try it yourself! Below is a Regents-style problem

Problem: Laura hits a golf ball from the ground. Its height in feet is modeled by −16t² + 48t, where t is time in seconds.

(a) When does the ball hit the ground?  (b) What is the maximum height?  (c) At what time does it reach maximum height?

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Click here to see the answer

1. (a) Set the expression equal to zero and factor:

   −16t² + 48t = 0

   
   

   −16t(t − 3) = 0

   
   

   t = 0 (when hit) or t = 3 seconds (when it lands)

2. (b) and (c) Axis of symmetry gives the time of max height:

   t = −48 / (2 × −16) = 48 / 32 = 1.5 seconds

3. Substitute t = 1.5 into the height expression:

   −16 × (1.5)² + 48 × 1.5 = −16 × 2.25 + 72 = −36 + 72 = 36 feet

Lands at t = 3 sec. Max height: 36 feet at t = 1.5 sec.

✗ COMMON TRAP

Reporting t = 3 as the time of max height. The zeros tell you when it's on the ground — the vertex tells you the max.

✓ THE RULE

Zero of the expression = on the ground. Vertex = highest point. One question about each.

For more practice with systems of quadratics, check out these resources:

https://momentofmath.com/mom.html? (look under "quadratics")