Given the expansion of $(1 + k_1x)(1 + k_2x)^6$, we are tasked with finding the values of $k_1$ and $k_2$ such that the coefficients of $x$ and $x^2$ are 8 and 30, respectively.

Step 1: Write the expansion for the first few terms

Expand $(1 + k_1x)(1 + k_2x)^6$ using the binomial theorem for $(1 + k_2x)^6$:

$(1 + k_2x)^6 = 1 + 6k_2x + 15k_2^2x^2 + \dots$

Now multiply this by $(1 + k_1x)$:

$(1 + k_1x)(1 + 6k_2x + 15k_2^2x^2 + \dots)$

Step 2: Collect terms involving $x$ and $x^2$

• The term involving $x$ comes from: $1 \cdot 6k_2x + k_1x \cdot 1 = (6k_2 + k_1)x$ The coefficient of $x$ is $6k_2 + k_1 = 8$.

• The term involving $x^2$ comes from: $1 \cdot 15k_2^2x^2 + k_1x \cdot 6k_2x = (15k_2^2 + 6k_1k_2)x^2$ The coefficient of $x^2$ is $15k_2^2 + 6k_1k_2 = 30$.

Step 3: Solve the system of equations

We now have two equations:

1. $6k_2 + k_1 = 8$
2. $15k_2^2 + 6k_1k_2 = 30$

From equation (1):

Solve for $k_1$: $k_1 = 8 - 6k_2$

Substitute this into equation (2):

$15k_2^2 + 6(8 - 6k_2)k_2 = 30$

Simplify: $15k_2^2 + 48k_2 - 36k_2^2 = 30$ $-21k_2^2 + 48k_2 - 30 = 0$

Divide through by -3 to simplify: $7k_2^2 - 16k_2 + 10 = 0$

Step 4: Solve the quadratic equation

Solve using the quadratic formula: $k_2 = \frac{-(-16) \pm \sqrt{(-16)^2 - 4(7)(10)}}{2(7)}$ $k_2 = \frac{16 \pm \sqrt{256 - 280}}{14}$ $k_2 = \frac{16 \pm \sqrt{-24}}{14}$

The discriminant is negative, indicating no real solution for $k_2$. Please recheck the values given or the setup of the problem.

Would you like me to explain this in more detail or check for any possible corrections?

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Here are 5 questions based on this problem:

1. How does the binomial theorem apply to expanding expressions like $(1 + k_2x)^6$?
2. Can you describe how coefficients of specific terms like $x$ or $x^2$ are identified during expansion?
3. Why is the quadratic formula used to solve for $k_2$, and what happens when there is a negative discriminant?
4. How would the process change if the discriminant were positive?
5. What are the implications of having no real solutions for this system of equations?

Tip: Always check the discriminant when solving quadratic equations to predict whether you'll get real, repeated, or complex roots.