Understanding Common Roots in Quadratic Equations

The problem asks us to find the values of b and c for two given quadratic equations, given that they share the exact same roots. The two equations are:

1. 4x2 + bx + 3 = 0
2. 8x2 + 4x + c = 0

A fundamental property of quadratic equations is that if two equations, say a1x2 + b1x + c1 = 0 and a2x2 + b2x + c2 = 0, have identical roots, then their coefficients must be proportional. This means there exists a constant ratio (let's call it k) such that:

$$ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = k $$

Calculating the Ratio of Coefficients

Let's identify the coefficients for our specific equations:

• For the first equation: a1 = 4, b1 = b, c1 = 3
• For the second equation: a2 = 8, b2 = 4, c2 = c

Now, we can set up the proportion using these coefficients:

$$ \frac{4}{8} = \frac{b}{4} = \frac{3}{c} $$

From the first part of the ratio, we can determine the constant proportionality factor:

$$ k = \frac{4}{8} = \frac{1}{2} $$

Finding the Value of b

We equate the middle term of the proportion to this ratio:

$$ \frac{b}{4} = \frac{1}{2} $$

To solve for b, we multiply both sides by 4:

$$ b = 4 \times \frac{1}{2} $$

$$ b = 2 $$

Determining the Value of c

Similarly, we equate the last term of the proportion to the ratio:

$$ \frac{3}{c} = \frac{1}{2} $$

To solve for c, we can cross-multiply or invert both sides:

$$ c = 3 \times 2 $$

$$ c = 6 $$

Conclusion

Therefore, the values of b and c are 2 and 6, respectively. This corresponds to the second option provided.