Subscribe to our YouTube channel for the latest videos, updates, and tips.


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 ax\(^{2}\) + 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 ax\(^{2}\) + bx + c = 0.

Therefore,

aα\(^{2}\) + bα + c = 0 ............... (i)

aβ\(^{2}\) + bβ + c = 0 ............... (ii)

aγ\(^{2}\) + bγ + c = 0 ............... (iii)

Subtracting (ii) from (i), we get

a(α\(^{2}\) - β\(^{2}\)) + 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(β\(^{2}\) - γ\(^{2}\)) + 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 ax\(^{2}\) + 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 3x\(^{2}\) - 4x - 4 = 0 by using the general expressions for the roots of a quadratic equation.

Solution:

The given equation is 3x\(^{2}\) - 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 = \(\frac{- b - \sqrt{b^{2} - 4ac}}{2a}\) and β = \(\frac{- b + \sqrt{b^{2} - 4ac}}{2a}\) we get

α = \(\frac{- (-4) - \sqrt{(-4)^{2} - 4(3)(-4)}}{2(3)}\) and β = \(\frac{- (-4) + \sqrt{(-4)^{2} - 4(3)(-4)}}{2(3)}\)

⇒ α = \(\frac{4 - \sqrt{16 + 48}}{6}\) and β =\(\frac{4 + \sqrt{16 + 48}}{6}\)

⇒ α = \(\frac{4 - \sqrt{64}}{6}\) and β =\(\frac{4 + \sqrt{64}}{6}\)

⇒ α = \(\frac{4 - 8}{6}\) and β =\(\frac{4 + 8}{6}\)

⇒ α = \(\frac{-4}{6}\) and β =\(\frac{12}{6}\)

⇒ α = -\(\frac{2}{3}\) and β = 2

Therefore, the roots of the given quadratic equation are 2 and -\(\frac{2}{3}\).

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



11 and 12 Grade Math 

From Quadratic Equation cannot have more than Two Roots to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Conversion of Improper Fractions into Mixed Fractions |Solved Examples

    May 12, 25 04:52 AM

    Conversion of Improper Fractions into Mixed Fractions
    In conversion of improper fractions into mixed fractions, we follow the following steps: Step I: Obtain the improper fraction. Step II: Divide the numerator by the denominator and obtain the quotient…

    Read More

  2. Multiplication Table of 6 | Read and Write the Table of 6 | Six Table

    May 12, 25 02:23 AM

    Multiplication Table of Six
    Repeated addition by 6’s means the multiplication table of 6. (i) When 6 bunches each having six bananas each. By repeated addition we can show 6 + 6 + 6 + 6 + 6 + 6 = 36 Then, six 6 times or 6 sixes

    Read More

  3. Word Problems on Decimals | Decimal Word Problems | Decimal Home Work

    May 11, 25 01:22 PM

    Word problems on decimals are solved here step by step. The product of two numbers is 42.63. If one number is 2.1, find the other. Solution: Product of two numbers = 42.63 One number = 2.1

    Read More

  4. Worksheet on Dividing Decimals | Huge Number of Decimal Division Prob

    May 11, 25 11:52 AM

    Worksheet on Dividing Decimals
    Practice the math questions given in the worksheet on dividing decimals. Divide the decimals to find the quotient, same like dividing whole numbers. This worksheet would be really good for the student…

    Read More

  5. Worksheet on Multiplying Decimals | Product of the Two Decimal Numbers

    May 11, 25 11:18 AM

    Practice the math questions given in the worksheet on multiplying decimals. Multiply the decimals to find the product of the two decimal numbers, same like multiplying whole numbers.

    Read More