The roots of the quadratic equation
$x^{2}-2\sqrt{3}x-22 = 0 $ are :
- imaginary
- real, rational and equal
- real, irrational and unequal
- real, rational and unequal
The Correct Option is C
Approach Solution - 1
Given, :
\(x^{2}-2 \sqrt{3} x-22=0\)
\(D=b^{2}-4 a c=12+4 \times 22=100>0\)
\(\therefore\) Roots are irrational, real and unequal.
So, the correct option is (C) : real, irrational and unequal.
Approach Solution -2
Discriminant D :
D = b2 - ac
Compare the given equation \(x^2-2\sqrt3-22=0\) with the standard equation i.e
\(ax^2+bx+c\) , so we get
⇒ \(a=1, b=-2\sqrt3,c=-22\)
By substituting these values into the discriminant, we obtain
\(⇒D=(-2\sqrt3)^2-4(1)(-22)\)
\(=12+88\)
\(=100\)
Now, nature of the root :
Since, D > 0 and is a perfect square. So the roots are real, rational and unequal.
So, the correct option is (C) : real, irrational and unequal.
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Concepts Used:
Quadratic Equations
A polynomial that has two roots or is of degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b, and c are the real numbers.
Consider the following equation ax²+bx+c=0, where a≠0 and a, b, and c are real coefficients.
The solution of a quadratic equation can be found using the formula, x=((-b±√(b²-4ac))/2a)
Two important points to keep in mind are:
- A polynomial equation has at least one root.
- A polynomial equation of degree ‘n’ has ‘n’ roots.
Read More: Nature of Roots of Quadratic Equation
There are basically four methods of solving quadratic equations. They are:
- Factoring
- Completing the square
- Using Quadratic Formula
- Taking the square root