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Important Points to Remember

The important concepts in the theory of equations are given below:

The general form of a quadratic equation in x is given by ax2 + bx + c = 0

The roots are given by x = (-b±√(b2 – 4ac))/2a

If α and β are the roots of the equation ax2 + bx + c = 0, a ≠ 0, then the sum of roots, α + β = -b/a.

Product of roots, αβ = c/a

If the sum and product of roots are known, then the quadratic equation is given by x2 – (sum of roots)x + product of roots = 0

For a quadratic equation, b2 – 4ac is known as the discriminant denoted by D.

If D = 0, the equation will have two equal real roots.

If D > 0, then the equation will have two distinct real roots.

If D < 0, then the equation has no real roots.

The graph of a quadratic equation is a parabola. The parabola will open upwards if a >0, and open downwards if a < 0.

If a > 0, when x = -b/2a, f(x) attains its minimum value.

If a < 0, when x = -b/2a, f(x) attains its maximum value.

Relationship between Roots and Coefficients

If α1, α2, α3, α4, α5, α6,…., αn are the roots of the quadratic equation:

­­­­­

Then, the sum of roots:

The sum of the product of roots taken two at a time:

The sum of the product of roots taken three at a time:

Product of Roots:

Therefore, for a cubic equation

Theory of Equation Solved Problems

Example 1: If one root of a cubic equation is double of another, then find all the roots of equation x3 + 36 = 7x2.

Solution:

Given, f (x) = x3 – 7x2 + 0.x + 36

Let α, β, and γ be the roots of the given cubic function f (x).

Therefore, α + β + γ = -b/a = 7 . . . . . . (1)

Also, α β + γ β + αγ = c/a = 0 . . . . . . . . . . (2)

And, αβγ = -d/a = – 36 . . . . . . . . (3)

Since, α = 2β [Given],

3β + γ = 7 [From Equation (1)] . . . . . . . . . (4)

Also, 2β2 + 3βγ = 0 [From Equation (2)]

β (2β + 3γ) = 0

Since, β ≠ 0, 2β + 3γ = 0 . . . . . . . . . . . . . (5)

And, 2β2 γ = – 36 [From Equation (3)] . . . . . . . . . . . . . . (6),

On solving Equation (4) and Equation (5), we get,

β = 3 and γ = – 2

Therefore, the roots of equation x3 + 36 = 7x2 are 3, 6, and – 2.

Answer:

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Step-by-step explanation:

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