Does there exist a quadratic equation whose coefficients are rational but both of its roots are irrational
Solution:
We have to determine if a quadratic equation whose coefficients are rational but both of its roots are irrational.
The standard form of a quadratic equation is ax2 + bx + c = 0 in variable x.
Where a, b, and c are real numbers and a ≠ 0.
A quadratic equation is an “equation of degree 2” that has two answers for x called the roots of the quadratic equations and are designated as (α, β).
For the roots to be irrational the discriminant should not be a perfect square.
✦ Try This: Consider the quadratic equation 2x² + x - 4 = 0, which has rational coefficients.
Given, the equation is 2x² + x - 4 = 0.
We have to find the roots of the equation.
Using the quadratic formula,
x = [-b ± √b² - 4ac]/2a
Here, a = 2, b = 1 and c = -4
x = [-1 ± √(1)² - 4(2)(-4)]/2(2)
x = [-1 ± √1 + 32]/ 4
x = [-1 ± √33]/4
The roots are [-1 + √33]/4 and [-1 - √33]/4
Therefore, the roots are irrational.
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 4
NCERT Exemplar Class 10 Maths Exercise 4.2 Problem 4
Does there exist a quadratic equation whose coefficients are rational but both of its roots are irrational
Summary:
The statement “a quadratic equation whose coefficients are rational but both of its roots are irrational” is true
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