Loading [MathJax]/jax/output/HTML-CSS/jax.js

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 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 

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. 8 Times Table | Multiplication Table of 8 | Read Eight Times Table

    May 18, 25 04:33 PM

    Printable eight times table
    In 8 times table we will memorize the multiplication table. Printable multiplication table is also available for the homeschoolers. 8 × 0 = 0 8 × 1 = 8 8 × 2 = 16 8 × 3 = 24 8 × 4 = 32 8 × 5 = 40

    Read More

  2. Worksheet on Average | Word Problem on Average | Questions on Average

    May 17, 25 05:37 PM

    In worksheet on average interest we will solve 10 different types of question. Find the average of first 10 prime numbers. The average height of a family of five is 150 cm. If the heights of 4 family

    Read More

  3. How to Find the Average in Math? | What Does Average Mean? |Definition

    May 17, 25 04:04 PM

    Average 2
    Average means a number which is between the largest and the smallest number. Average can be calculated only for similar quantities and not for dissimilar quantities.

    Read More

  4. Problems Based on Average | Word Problems |Calculating Arithmetic Mean

    May 17, 25 03:47 PM

    Here we will learn to solve the three important types of word problems based on average. The questions are mainly based on average or mean, weighted average and average speed.

    Read More

  5. Rounding Decimals | How to Round a Decimal? | Rounding off Decimal

    May 16, 25 11:13 AM

    Round off to Nearest One
    Rounding decimals are frequently used in our daily life mainly for calculating the cost of the items. In mathematics rounding off decimal is a technique used to estimate or to find the approximate

    Read More