CCSS.MATH.HSF.TF.A.1 / AP.CALC.LIM.1✓ New Hampshire Department of Education AlignedAcademic Year 2026–27
Quadratic Functions & The Quadratic Formula
Derive and apply the quadratic formula $x = (-b \pm \sqrt{b^2 - 4ac}) / 2a$ to find real and complex roots of second-degree polynomials.
Why It Matters:Models projectile trajectories in astrophysics, satellite parabolic antennas, and economic revenue curves.
Curated Video Instruction
● Verified Active EmbedSTEP-BY-STEP PROOF
Worked Example
Find roots of 2x^2 + 5x - 3 = 0 using quadratic formula with a=2, b=5, c=-3:
1
Calculate discriminant b^2 - 4ac
5^2 - 4(2)(-3) = 25 - (-24) = 49
Positive discriminant 49 indicates two distinct rational roots.
2
Compute square root of discriminant
sqrt(49) = 7
Square root of 49 is 7.
3
Evaluate two roots
x = (-5 + 7)/4 = 2/4 = 0.5 and x = (-5 - 7)/4 = -12/4 = -3
Roots are x = 1/2 and x = -3.
✓Verification Check
Verify root x = -3: 2(-3)^2 + 5(-3) - 3 = 2(9) - 15 - 3 = 18 - 18 = 0.
18 - 18 = 0
•
0 = 0
⚠️ Common Pitfalls & Misconceptions
Sign error with negative c value in discriminant
✗ Calculating 25 - 24 = 1 because forgetting that -4(2)(-3) is +24✓ Multiplying two negative factors yields a positive: -4 * 2 * (-3) = +24
Two negative signs multiply to a positive.
INTERACTIVE DRILLS
Guided Practice
Practice Question1 of 1
Step-by-step Feedback
What does a discriminant of zero (b^2 - 4ac = 0) signify?
MASTERY BENCHMARK
Official Assessment Quiz
Formal Assessment
Quadratic Functions & The Quadratic Formula
Passing Criteria:70% or higher
Question 1 of 1
What is the y-intercept of f(x) = 3x^2 - 4x + 7?
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Where This Lesson Leads
Ballistics Specialists, Structural Engineers, and Quantitative Analysts apply quadratic functions daily.