What is the quadratic formula?+
The quadratic formula solves ax² + bx + c = 0: x = (−b ± √(b²−4ac)) / (2a). It is derived by completing the square on the general form. The ± produces two solutions. If the discriminant D = b²−4ac is negative, the roots are complex conjugates: x = −b/(2a) ± i√(|D|)/(2a).
What is the discriminant and what does it tell you?+
D = b² − 4ac. If D > 0: two distinct real roots (parabola crosses x-axis twice). If D = 0: one repeated real root (parabola tangent to x-axis). If D < 0: two complex conjugate roots (parabola does not touch x-axis). The discriminant determines the nature of roots before computing them.
How do I factor a quadratic equation?+
For x² + bx + c, find two numbers that multiply to c and add to b. Example: x² + 5x + 6 = (x+2)(x+3) because 2×3=6 and 2+3=5. For ax² + bx + c with a≠1, use the AC method: find factors of ac that add to b, then split the middle term. Factoring is only possible over integers when D is a perfect square.
What is completing the square?+
Transform ax² + bx + c into a(x + h)² + k form. Steps: 1) divide by a; 2) move c to the right; 3) add (b/2a)² to both sides; 4) write the left side as a perfect square. This reveals the vertex (−h, k) of the parabola and derives the quadratic formula. Example: x²+4x+1=0 → (x+2)²=3 → x=−2±√3.
What are Vieta's formulas?+
For roots r₁ and r₂ of ax²+bx+c=0: r₁+r₂ = −b/a and r₁×r₂ = c/a. These hold without actually computing the roots. Example: x²−5x+6=0 → r₁+r₂=5, r₁×r₂=6, confirming roots 2 and 3. Vieta's formulas extend to higher-degree polynomials and are used in symmetric function theory and coding theory.
What does the vertex of a parabola represent?+
The vertex is the turning point: maximum if a < 0, minimum if a > 0. x-coordinate of vertex: −b/(2a). y-coordinate: c − b²/(4a), or substitute x back into the equation. In optimization problems, the vertex gives the optimal value — e.g., the price maximizing revenue, the angle maximizing projectile range.
How do complex roots arise in quadratic equations?+
When D = b²−4ac < 0, √D = i√|D| where i = √(−1). The roots are x = −b/(2a) ± i√(|D|)/(2a), a complex conjugate pair a ± bi. Complex roots always appear in conjugate pairs for real-coefficient polynomials. They represent the equation having no real x-intercepts but are essential in electrical engineering (impedance) and signal processing.
What is the difference between roots, solutions, and zeros?+
All three refer to values of x satisfying the equation: roots = solutions = zeros. "Zero of a function" means f(x) = 0; "root of an equation" means the equation equals zero; "solution" is the most general term. For the quadratic ax²+bx+c=0, roots are the x-values making the expression zero. The parabola y=ax²+bx+c crosses the x-axis at the real roots.
Can I solve a quadratic without the formula?+
Yes. Three alternatives: 1) Factoring (fastest when roots are simple integers). 2) Completing the square (gives vertex form too). 3) Graphing (approximate). 4) Numerically (Newton-Raphson iteration). The quadratic formula is the universal method — it always works and gives exact answers. For repeated use, the formula is most efficient.
What is a perfect square trinomial?+
A trinomial a²x²+2abx+b² that factors as (ax+b)². Recognizing it speeds up solving: x²+6x+9=(x+3)². The discriminant of a perfect square trinomial equals 0, confirming one repeated root. Example: 4x²−12x+9=(2x−3)², root x=3/2 (repeated). Completing the square transforms any quadratic into this form.
How are quadratics used in projectile motion?+
Height h(t) = h₀ + v₀t − ½gt², where g = 9.8 m/s². This is a quadratic in t. The vertex gives maximum height at t = v₀/g. Setting h(t) = 0 and solving the quadratic gives the time of flight. Example: ball thrown upward at 20 m/s from 2m height: 0 = 2 + 20t − 4.9t². D = 400 + 4(4.9)(2) = 439.2. Time of flight ≈ (−20 − √439.2)/(−9.8) ≈ 4.18 s.
What is the sum and product of roots formula for higher-degree polynomials?+
For the quadratic, Vieta's formulas are r₁+r₂=−b/a and r₁r₂=c/a. For a cubic x³+px²+qx+r=0 with roots r₁,r₂,r₃: sum=−p, sum of products in pairs=q, product=−r. Generally, the k-th elementary symmetric polynomial of n roots equals (−1)^k × (coefficient of x^(n−k)) / (leading coefficient). Used in coding theory, cryptography, and polynomial factorization.