Question

Heya everyone.
Prove that - "A quadratic equation cannot have more than 2 roots."​

Answer 1

Answer:

Let us prove a quadratic equation cannot have more than 2 roots. :

Let us have a quadratic equation f(x) so 0 will

f(x)=ax2+bx+c = a not equal to 0

quadratic equation cannot have more than 2 roots ,

then factories f(x) so the results are

a(x−a1)(x−a2)(x−a3)

hence, a1,a2,a3 are roots of the equation f(x)=0

hence Then each α,β and γ will satisfy this quadratic equation.

hope it helps

thanks

Answer 2

Let's strat from the very beginning.

We need to define some terms before proving "A quadratic equation cannot have more than 2 roots".

☯︎ Equation

Equation are algebraic expressions that consist of variables and coefficients.

☯︎ Degree of a equation

The highest power of the variable in an equation is called Degree of the Equation.

✳️ Number of roots of any equation

The number of roots of any equation is the same as that of its degree.

If there's an equation of power ‘n’. Let's say p(x) = axⁿ + b (a ≠ 0).Then the number of roots of the equation is ‘n’.

✴️ Types of Equation

There are many types of equations depending on the degree of the equation.

❶ Linear Equation

Linear Equation has degree one.

General Form: p(x) = ax + b  (a ≠ 0)

Number of roots = 1

Example: 7x + 4 = 9 is Linear Equation in x.

❷ Quadratic Equation

Quadratic Equation has degree two.

General Form p(x) = ax² + bx + c (a ≠ 0)

Number of roots = 2

Example: 7x² + 5x + 2 = 14 is a Quadratic Equation in x.

❸ Cubic Equation

Cubic Equation has degree three.

General Form p(x) = ax³ + bx² + cx + d (a ≠ 0)

Number of roots = 3

Example: 7x³ + 5x² + 6x + 5 = 14  is a Cubic Equation in x.

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Now coming to your question.

You've asked me to prove that "A quadratic equation cannot have more than 2 roots".

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❖ Method 1

General Form of Quadratic Equation p(x) = ax² + bx + c (a ≠ 0)

The highest power of the quadratic equation is two and we know that the highest number of roots that an equation can have is same as the power of the equation. So the utmost number of roots a quadratic equation can have is two because the degree of quadratic equation is two.

You can prove in the same manner for other equations too like cubic , biquadratic and so on.

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❖ Method 2

Let's take any quadratic equation as

p(x) = ax² + bx + c (a ≠ 0)

Let the roots of the quadratic equation be α , β and γ respectively.

By factor theorem (x - α) , (x - β) and (x - γ) are the factors of p(x).

k[(x - α) (x - β) (x - γ)]      [For some constant k]

= k{x³ - (α + β + γ)x² + (αβ + βγ + γα)x - (αβγ)}

= kx³ - k(α + β + γ)x² + k(αβ + βγ + γα)x - k(αβγ)

When we factorise the polynomial we find the degree of the polynomial is 3 whereas the degree of the unfactorised polynomial is two. So they can never ever be equal. So we conclude that the p(x) can't have 3 roots.

You can prove in the same manner for other equations too like cubic , biquadratic and so on.

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