Quadratic Equations
Quadratic Equations tests standard form, solution methods, the discriminant, and applications. A quadratic equation is ax^2+bx+c=0 with a ≠ 0.
Core rules
- Factorisation works when the middle term splits using factors whose product is ac and sum is b.
- Completing the square rewrites the equation as (x + b/2a)^2 = (b^2 - 4ac)/4a^2.
- Quadratic formula: x = (-b ± sqrt(b^2 - 4ac))/(2a).
- Discriminant D = b^2 - 4ac. If D > 0, two distinct real roots; D = 0, two equal real roots; D < 0, no real roots.
- For real roots, sum = -b/a and product = c/a.
Worked example
Solve 2x^2 - 7x + 3 = 0. Here a=2, b=-7, c=3, so D=49-24=25. Since D>0, roots are real and distinct. x=(7 ± 5)/4, giving x=12/4=3 and x=2/4=1/2. Check: 2(3)^2-7(3)+3=18-21+3=0 and 2(1/2)^2-7(1/2)+3=0.5-3.5+3=0.
Traps
Do not drop the negative sign in -b. The formula requires a ≠ 0. D<0 means no real roots, not no roots at all. In applications, reject answers that violate the context, such as a negative length, even if they satisfy the equation.