Does there exist a quadratic equation whose coefficients are all distinct irrationals but both the roots are rationals
Solution:
We have to determine if a quadratic equation whose coefficients are irrational 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).
✦ Try This: Consider the equation √3x² - √27x + √12 = 0.
Given, the equation is √3x² - √27x + √12 = 0.
We have to determine the roots of the equation.
Using the quadratic formula,
x = [-b ± √b² - 4ac]/2a
Here, a = √3, b = -√27 and c = √12
\(x=\frac{-(-\sqrt{27})\pm \sqrt{(-\sqrt{27})^{2}-4(\sqrt{3})(\sqrt{12})}}{2(\sqrt{3})}\\=\frac{\sqrt{27}\pm \sqrt{27-4\sqrt{36}}}{2\sqrt{3}}\\=\frac{\sqrt{27}\pm \sqrt{27-24}}{2\sqrt{3}}\\=\frac{3\sqrt{3}\pm \sqrt{3}}{2\sqrt{3}}\\=\frac{3\pm 1}{2}\)
Now, x = (3 + 1)/2 = 4/2 = 2
x = (3 - 1)/2 = 2/2 = 1
The roots of the equation are 1 and 2.
Therefore, the equation with irrational coefficients can have 2 rational roots
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 4
NCERT Exemplar Class 10 Maths Exercise 4.2 Problem 5
Does there exist a quadratic equation whose coefficients are all distinct irrationals but both the roots are rationals
Summary:
The statement “a quadratic equation whose coefficients are all distinct irrationals but both the roots are rationals” is true
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