Quadratic Equation cannot have more than Two Roots

We will discuss here that a quadratic equation cannot have more than two roots.

Proof:

Let us assumed that α, β and γ be three distinct roots of the quadratic equation of the general form ax2 + bx + c = 0, where a, b, c are three real numbers and a ≠ 0. Then, each one of α, β and γ will satisfy the given equation ax2 + bx + c = 0.

Therefore,

2 + bα + c = 0 ............... (i)

2 + bβ + c = 0 ............... (ii)

2 + bγ + c = 0 ............... (iii)

Subtracting (ii) from (i), we get

a(α2 - β2) + b(α - β) = 0

⇒ (α - β)[a(α + β) + b] = 0

⇒ a(α + β) + b = 0, ............... (iv) [Since, α and β are distinct, Therefore, (α - β) ≠ 0]

Similarly, Subtracting (iii) from (ii), we get

a(β2 - γ2) + b(β - γ) = 0

⇒ (β - γ)[a(β + γ) + b] = 0

⇒ a(β + γ) + b = 0, ............... (v) [Since, β and γ are distinct, Therefore, (β - γ) ≠ 0]

Again subtracting (v) from (iv), we get

a(α - γ) = 0

⇒ either a = 0 or, (α - γ) = 0

But this is not possible, because by the hypothesis a ≠ 0 and α - γ ≠ 0 since α ≠ γ

α and γ are distinct.

Thus, a(α - γ) = 0 cannot be true.

Therefore, our assumption that a quadratic equation has three distinct real roots is wrong.

Hence, every quadratic equation cannot have more than 2 roots.

 

Note: If a condition in the form of a quadratic equation is satisfied by more than two values of the unknown then the condition represents an identity.

Consider the quadratic equation of the general from ax2 + bx + c = 0 (a ≠ 0) ............... (i)


Solved examples to find that a quadratic equation cannot have more than two distinct roots

Solve the quadratic equation 3x2 - 4x - 4 = 0 by using the general expressions for the roots of a quadratic equation.

Solution:

The given equation is 3x2 - 4x - 4 = 0

Comparing the given equation with the general form of the quadratic equation ax^2 + bx + c = 0, we get

a = 3; b = -4 and c = -4

Substituting the values of a, b and c in α = bb24ac2a and β = b+b24ac2a we get

α = (4)(4)24(3)(4)2(3) and β = (4)+(4)24(3)(4)2(3)

⇒ α = 416+486 and β =4+16+486

⇒ α = 4646 and β =4+646

⇒ α = 486 and β =4+86

⇒ α = 46 and β =126

⇒ α = -23 and β = 2

Therefore, the roots of the given quadratic equation are 2 and -23.

Hence, a quadratic equation cannot have more than two distinct roots.



11 and 12 Grade Math 

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