Quadratic Equations Questions and Answers

Explanation :

First, we have to find the discriminant of the Quadratic Equation :
Discriminant = b^{2}-4ac

Here, a = 1,b = 5 and c = 6
So, Discriminant = 1 > 0
So, the roots are real and have two distinct roots.
Hence Option (D) is correct.

To calculate positive roots of Eqn. f(a) =  a^{4}+ a^{2}+1= 0

Since all the signs are positive, therefore none of the roots will be positive.

To calculate the negative roots of Eqn. f(a)= a^{4}+ a^{2}+1= 0

f(-a)= (- a)^{4} (- a)^{2}+1= 0

Again, as all the signs are same, hence no roots of this eqn. will be negative and also none of the roots will be equal to 0

Therefore Option D is the correct one.

Since f(0)<0 and f(1)>0, therefore,the root of the equation will be somewhere between 0 and 1

Now for all the negative values of p<0, the above equation will give negative values as p11 will eventually convert to a significant positive number. 

Therefore it will not have any real root for p>1 or p<0 

Hence the total number of roots= 1 

Thus option B is the correct one. 

α+β=p+6
αβ=2(2p-1)
According to the question,
p+6 = \frac{2(2p-1)}{2}

=> p = 7.

Option A – has no real roots

Option B is independent of term x²

Option C is also independent of term x² for k= -1

Option D implies

x² =1

x= ±1

Hence answer is Option D

Explanation:

To solve the inequality, we first find the roots of the quadratic equation 2x^2 + 5x - 3 = 0.

Using the quadratic formula, we get

x = ( -5 ± √(5^2 - 4 * 2 * -3) ) / (2 * 2)

=> x = ( -5 ± √49 ) / 4.

The roots are x = ( -5 + 7 ) / 4 = 1/2 and x = ( -5 - 7 ) / 4 = -3.

We plot these values on a number line and test a value in each interval to check if it satisfies the inequality.

For example, testing x = 0, we find 2(0)^2 + 5(0) - 3 = -3 < 0.

Thus, the solution is x > 1/2 and x < -3.

The correct answer is option D.

Sum of all the roots 3a^{2} +2a-1 = (−x/y) = −2/3

Hence Option A is the correct one.