Answer
Verified
387.9k+ views
Hint: To solve the above question, we have to know the concept of quadratic equation. Let we consider a quadratic equation as $A{{x}^{2}}+Bx+C=0$ where the discriminant is ${{B}^{2}}-4AC$ .We have to know if in any quadratic equation, coefficient of ${{x}^{2}}$ and constant term are of opposite sign$\left( AC<0 \right)$ , then that equation will definitely have real roots.
Complete step by step answer:
Now given that,
$P\left( x \right)=a{{x}^{2}}+bx+c$
And $q(x)=-a{{x}^{2}}+dx+c=0$
Now we will consider the equation, $A{{x}^{2}}+Bx+C=0$ where if we find the value of $x$
Then we use the formula of $\dfrac{-B\pm \sqrt{{{B}^{2}}-4AC}}{2A}$ .
Here the discriminant is ${{B}^{2}}-4AC$ .
We can see that in the above equation $A$ and $C$ are of the opposite sign that is $AC$ is negative. So, we can see that ${{B}^{2}}-4AC$ will be positive and hence we also see that the equation will have real roots.
So, we have to know if in any quadratic equation, coefficient of ${{x}^{2}}$ and constant term are of opposite sign$\left( AC<0 \right)$ , then that Equation will definitely have real roots.
Now we take the $P\left( x \right)=a{{x}^{2}}+bx+c$
Where we can see that product of coefficient of ${{x}^{2}}$ and constant term $=ac$
And for $q(x)=-a{{x}^{2}}+dx+c=0$
Where we can see that product of coefficient of ${{x}^{2}}$ and constant term $=-ac$
So, we can see that if $ac$ is positive, then $-ac$ will be negative and vice versa.
And for this case one of the equations will definitely have real roots.
That is $\left( a{{x}^{2}}+bx+c \right)\left( -a{{x}^{2}}+dx+c \right)=0$ will have at least two real roots.
So, the correct answer is “Option B”.
Note: Here student must take care of the concept of discriminant and also take care of if coefficient of ${{x}^{2}}$ and constant term are of opposite sign$\left( AC<0 \right)$ , then that Equation will definitely have real roots.
Sometimes, a student makes a mistake by using the discriminant formula that is ${{B}^{2}}-4AC$. So, Students have to take care of it.
Complete step by step answer:
Now given that,
$P\left( x \right)=a{{x}^{2}}+bx+c$
And $q(x)=-a{{x}^{2}}+dx+c=0$
Now we will consider the equation, $A{{x}^{2}}+Bx+C=0$ where if we find the value of $x$
Then we use the formula of $\dfrac{-B\pm \sqrt{{{B}^{2}}-4AC}}{2A}$ .
Here the discriminant is ${{B}^{2}}-4AC$ .
We can see that in the above equation $A$ and $C$ are of the opposite sign that is $AC$ is negative. So, we can see that ${{B}^{2}}-4AC$ will be positive and hence we also see that the equation will have real roots.
So, we have to know if in any quadratic equation, coefficient of ${{x}^{2}}$ and constant term are of opposite sign$\left( AC<0 \right)$ , then that Equation will definitely have real roots.
Now we take the $P\left( x \right)=a{{x}^{2}}+bx+c$
Where we can see that product of coefficient of ${{x}^{2}}$ and constant term $=ac$
And for $q(x)=-a{{x}^{2}}+dx+c=0$
Where we can see that product of coefficient of ${{x}^{2}}$ and constant term $=-ac$
So, we can see that if $ac$ is positive, then $-ac$ will be negative and vice versa.
And for this case one of the equations will definitely have real roots.
That is $\left( a{{x}^{2}}+bx+c \right)\left( -a{{x}^{2}}+dx+c \right)=0$ will have at least two real roots.
So, the correct answer is “Option B”.
Note: Here student must take care of the concept of discriminant and also take care of if coefficient of ${{x}^{2}}$ and constant term are of opposite sign$\left( AC<0 \right)$ , then that Equation will definitely have real roots.
Sometimes, a student makes a mistake by using the discriminant formula that is ${{B}^{2}}-4AC$. So, Students have to take care of it.
Recently Updated Pages
Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred
The branch of science which deals with nature and natural class 10 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Define absolute refractive index of a medium
Find out what do the algal bloom and redtides sign class 10 biology CBSE
Prove that the function fleft x right xn is continuous class 12 maths CBSE
Trending doubts
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE
Which type of bond is stronger ionic or covalent class 12 chemistry CBSE
What organs are located on the left side of your body class 11 biology CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
How fast is 60 miles per hour in kilometres per ho class 10 maths CBSE
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
When people say No pun intended what does that mea class 8 english CBSE