Quadratic Equation Part 1 Quiz

x² - 5x + 6 = 0
or, (x - 2)(x - 3) = 0
So the roots of the equation are 2 and 3

Discriminant of a quadratic equation ax² + bx + c = 0 is given by:
D = (b² - 4ac)
For the given equation the discriminant is:
D = (-4)² - 4(2)(3) = -8

If α, β are roots of the quadratic equation ax² + bx + c = 0 then sum of the roots (α + β) = -b/a and product of the roots αβ = c/a
For the given equation αβ = c/a = 10 and α = 5, so the other root β = 10/5 = 2

If α, β are roots of the quadratic equation ax² + bx + c = 0 then sum of the roots (α + β) = -b/a
For the given equation (α + β) = -b/a = -(-5)/(2) = 5/2

If α, β are roots of the quadratic equation ax² + bx + c = 0 then product of the roots αβ = c/a
For the given equation αβ = c/a = (-8)/3 = -8/3

When D=0, the quadratic has two equal real roots.

For the given equation the discriminant is:
D = (2)² - 4(1)(5) = -16
Since D < 0 the roots are complex in nature

If α, β are roots of a quadratic equation then the equation is x² - (α + β)x + αβ = 0.
So the equation is:
x² - (8)x + 15 = 0
or, x² - 8x + 15 = 0

Substituting (-2) in x² + kx + 6 = 0 we get
(-2)² + k(-2) + 6 = 0
or, 4 - 2k + 6 = 0
or, k = 5

A negative discriminant means the quadratic has no real solutions, only complex conjugates.