Solving Quadratic Equations with Common Roots

This problem involves finding the value of k given two quadratic equations that share identical roots. We need to utilize the properties of quadratic equations when they have common roots.

Condition for Common Roots

If two quadratic equations, represented as $a_1x^2 + b_1x + c_1 = 0$ and $a_2x^2 + b_2x + c_2 = 0$, have both roots in common, then the ratio of their corresponding coefficients must be equal. This condition can be expressed as:

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

Applying the Condition to the Given Equations

The two given quadratic equations are:

• Equation 1: $2x^2 + 8x + 6 = 0$
• Equation 2: $kx^2 + 4kx + 3 = 0$

Let's identify the coefficients for each equation:

Now, we apply the condition for common roots:

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

Substituting the coefficients:

$$ \frac{2}{k} = \frac{8}{4k} = \frac{6}{3} $$

Calculating the Value of k

We can simplify the ratios and solve for k. Let's look at the ratios:

1. The first ratio is $\frac{2}{k}$.
2. The second ratio is $\frac{8}{4k}$. Simplifying this gives $\frac{8 \div 4}{4k \div 4} = \frac{2}{k}$. This confirms consistency between the coefficients of $x^2$ and $x$.
3. The third ratio is $\frac{6}{3}$. Simplifying this gives $2$.

Since both roots are common, all these ratios must be equal. We can equate the first ratio (or the second) to the third ratio:

$$ \frac{2}{k} = 2 $$

To solve for k, we can multiply both sides by k:

$$ 2 = 2k $$

Now, divide both sides by 2:

$$ k = \frac{2}{2} $$

$$ k = 1 $$

Therefore, the value of k for which both quadratic equations have common roots is 1.