Answer
Verified
480.3k+ views
Hint: Recollect the quadratic formula which is used to find the roots of a quadratic equation. This quadratic formula consists of a term which is discriminant in it. Assume the discriminant < 0 and try to find the nature of roots of the quadratic equation.
Before proceeding with the question, we must know the formulas that will be required to solve this question.
In quadratic equations, we have quadratic formula from which, the roots of the equation $a{{x}^{2}}+bx+c=0$ are given by,
$x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
In this formula, the term ${{b}^{2}}-4ac$ is the discriminant because discriminant $D={{b}^{2}}-4ac$. Substituting ${{b}^{2}}-4ac=D$ in the above equation, we get,
$x=\dfrac{-b\pm \sqrt{D}}{2a}...................\left( 1 \right)$
From the above equation, we can find the nature of roots for different values of discriminant $D$.
In the question, it is given that the discriminant $D$ for a quadratic equation is less than 0 and we have to find the nature of the roots.
Since it is given that the discriminant is less than 0, let us put substitute D<0 in equation $\left( 1 \right)$. From the complex numbers, we know that the square root of a negative number is an imaginary number. In equation $\left( 1 \right)$, D is negative and it is inside the square root. So, we can say that x is an imaginary number i.e. x is not a real number. So, the real roots of the quadratic equation do not exist.
Hence, the answer is option (a).
Note: To solve this question in a lesser time, one should remember it as a fact that any quadratic equation with discriminant less than 0 does not have real roots. One should also remember that for discriminant = 0, the quadratic equation has real and equal roots and for discriminant greater than 0, the quadratic equation has real and distinct roots.
Before proceeding with the question, we must know the formulas that will be required to solve this question.
In quadratic equations, we have quadratic formula from which, the roots of the equation $a{{x}^{2}}+bx+c=0$ are given by,
$x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
In this formula, the term ${{b}^{2}}-4ac$ is the discriminant because discriminant $D={{b}^{2}}-4ac$. Substituting ${{b}^{2}}-4ac=D$ in the above equation, we get,
$x=\dfrac{-b\pm \sqrt{D}}{2a}...................\left( 1 \right)$
From the above equation, we can find the nature of roots for different values of discriminant $D$.
In the question, it is given that the discriminant $D$ for a quadratic equation is less than 0 and we have to find the nature of the roots.
Since it is given that the discriminant is less than 0, let us put substitute D<0 in equation $\left( 1 \right)$. From the complex numbers, we know that the square root of a negative number is an imaginary number. In equation $\left( 1 \right)$, D is negative and it is inside the square root. So, we can say that x is an imaginary number i.e. x is not a real number. So, the real roots of the quadratic equation do not exist.
Hence, the answer is option (a).
Note: To solve this question in a lesser time, one should remember it as a fact that any quadratic equation with discriminant less than 0 does not have real roots. One should also remember that for discriminant = 0, the quadratic equation has real and equal roots and for discriminant greater than 0, the quadratic equation has real and distinct roots.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE