Quadratic Lab · Roots & Discriminant
Move the coefficients of a parabola and watch its vertex, axis, discriminant and real roots change in real time.
NCERT alignment: Class 11; chapter Complex Numbers and Quadratic Equations. Exam practice: JEE Main, JEE Advanced.
Learning objectives
Students should use the live simulator to test a quantitative prediction, read measurements and explain the limits of the model. Do not rely on the animation alone: calculate the expected result before adjusting controls.
Theory, equations and interpretations
Standard form and roots
Real roots are the x-intercepts of y=ax²+bx+c. The quadratic formula determines every root, including complex roots.
Important relation: ax² + bx + c = 0; a ≠ 0
Discriminant
If Δ>0 there are two distinct real roots; Δ=0 gives one repeated real root; Δ<0 gives two non-real complex conjugate roots.
Important relation: Δ = b² − 4ac; x = (−b ± √Δ)/(2a)
Vertex and symmetry
Complete the square: f(x)=a(x−xᵥ)²+yᵥ. The axis is x=xᵥ. The vertex is a minimum when a>0, maximum when a<0.
Important relation: xᵥ = −b/(2a); yᵥ = −Δ/(4a)
Parameter sensitivity
Change b to shift the vertex horizontally and c to shift the graph vertically. Compare numerical probes with the plotted curve.
Important relation: f′(x)=2ax+b
Practical investigations
- Predict the roots: Set a=1,b=−2,c=−3. Solve the equation by factorisation, then compare x-intercepts on screen.
- Collapse the intersections: Use a=1,b=−2,c=1. Explain why the parabola only touches the x-axis.
- No real roots: Change to a=1,b=0,c=2; explain why roots are complex even though the curve is visible.
- Reflect the graph: Change the orientation, hold b=c=0 and examine the maximum/minimum change.
Worked exam-style questions and answers
Q1. For x²−5x+6=0, the roots are
Options: A. 2 and 3; B. −2 and −3; C. 1 and 6; D. −1 and −6
Answer: A. Factorise (x−2)(x−3)=0.
Q2. For x²+4x+4=0, Δ equals
Options: A. 16; B. 0; C. −16; D. 4
Answer: B. Δ=4²−4×1×4=0, so the root −2 repeats.
Q3. If Δ<0 for a real quadratic, then
Options: A. two real roots; B. one real double root; C. no real roots; D. infinitely many roots
Answer: C. Negative discriminant gives non-real conjugate roots.
Q4. Vertex x-coordinate of 2x²+8x+1 is
Options: A. −4; B. −2; C. 2; D. 4
Answer: B. xᵥ=−b/(2a)=−8/4=−2.
Q5. For a<0 the quadratic vertex is
Options: A. minimum; B. inflection point; C. maximum; D. always at origin
Answer: C. A downward opening parabola has a maximum at its vertex.
Q6. How many real intercepts has x²+1=0?
Options: A. 0; B. 1; C. 2; D. infinitely many
Answer: A. Δ=−4, hence no real x-intercepts.
Common misconceptions and limitations
Model is y=ax²+bx+c with a≠0. Negative discriminant means no real x-axis intersections; complex roots exist algebraically but are not plotted as real points.
All learning content is available free of charge. The simulation is a simplified educational model. Check assumptions and units before applying a result. NANDAX SimStudio is not affiliated with NCERT or examination boards.
Teacher classroom use
Ask students to form a hypothesis, vary one parameter at a time, keep a record of graph readings, explain any change and compare results with the analytic relation. The on-screen worksheet offers questions with an answer key.