If the equations 2x² 7x+3=0 and 4x²+ax-3=0 have a common root, then what is the value of a?

Solution:

Given: The equations are: 1. 2x² + 7x + 3 = 0  and  2. 4x² + ax - 3 = 0

It is mentioned that both equations have a common root.

We need to determine the value of a. 

Concept Used: When two quadratic equations have a common root, we can equate the value of the root from one equation to the other by substituting it back into the second equation.

The generic quadratic equation is: ax² + bx + c = 0

The sum and product of roots are given by:

1. Sum of roots = -b / a and 2. Product of roots = c / a

For a common root between two equations, substitute the root from the first equation into the second equation and solve for the unknown parameter.

Calculation:

Step 1: Solve the first equation for its roots using the quadratic formula:  

For 2x² + 7x + 3 = 0, the quadratic formula is: x = (-b ± √(b² - 4ac)) / 2a

Here, a = 2, b = 7, c = 3.

⇒ x = (-7 ± √(7² - 4 × 2 × 3)) / (2 × 2)

⇒ x = (-7 ± √(49 - 24)) / 4

⇒ x = (-7 ± √25) / 4 ⇒ x = (-7 ± 5) / 4  

So, the roots are: x = (-7 + 5) / 4 = -2 / 4 = -0.5 and x = (-7 - 5) / 4 = -12 / 4 = -3

Step 2: Use the common root condition:

Let the common root be x = -3 (or x = -0.5).

Substitute x = -3 into the second equation, 4x² + ax - 3 = 0.

Substitute x = -3:   4(-3)² + a(-3) - 3 = 0 ⇒ 4(9) - 3a - 3 = 0 ⇒ 36 - 3a - 3 = 0 ⇒ 33 = 3a ⇒ a = 33 / 3 ⇒ a = 11  

Substitute x = -0.5:   4(-0.5)² + a(-0.5) - 3 = 0 ⇒ 4(0.25) - 0.5a - 3 = 0 ⇒ 1 - 0.5a - 3 = 0 ⇒ -2 = 0.5a ⇒ a = -2 / 0.5 ⇒ a = -4

Conclusion: The value of a is either 11 or -4.

∴ Correct Answer: Option 1: 11 or -4