2x²-14x+3k-4=0, dan a²-b² =21, maka nilai k adalah…

Given the equation \( 2x^2 - 14x + 3k - 4 = 0 \) and the equation \( a^2 - b^2 = 21 \), we need to find the value of \( k \).

### Step 1: Solve the quadratic equation for discriminant
A quadratic equation is generally in the form \( ax^2 + bx + c = 0 \). Here, the discriminant \( \Delta \) is given by:

\[
\Delta = b^2 - 4ac
\]

For the quadratic equation \( 2x^2 - 14x + (3k - 4) = 0 \), we have:

- \( a = 2 \)
- \( b = -14 \)
- \( c = 3k - 4 \)

So, the discriminant is:

\[
\Delta = (-14)^2 - 4(2)(3k - 4)
\]
\[
\Delta = 196 - 8(3k - 4)
\]
\[
\Delta = 196 - 24k + 32
\]
\[
\Delta = 228 - 24k
\]

### Step 2: Use the given condition
The condition \( a^2 - b^2 = 21 \) implies that the quadratic equation must have two distinct real roots. For this to happen, the discriminant must be greater than zero:

\[
228 - 24k > 0
\]
\[
228 > 24k
\]
\[
k < \frac{228}{24} = 9.5
\]

### Step 3: Analyze further using \( a^2 - b^2 = 21 \)
To satisfy \( a^2 - b^2 = 21 \) for the roots \( a \) and \( b \), this means the discriminant must be related to the equation in a specific manner (considering the nature of roots). However, without additional specifics about \( a \) and \( b \), the quadratic discriminant's condition is what we primarily use.

### Conclusion
\( k \) must be less than 9.5 to satisfy the discriminant condition. However, the exact value of \( k \) depends on additional specific values or relations between \( a \) and \( b \). 

If more conditions or constraints are given, we could refine this value further. 

Would you like to clarify or provide additional conditions? If not, do you have any questions?

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Here are 5 related questions:

1. How do we determine the roots of a quadratic equation using the discriminant?
2. What is the significance of the discriminant being greater than zero?
3. Can \( a^2 - b^2 = 21 \) give us more specific information about the roots?
4. How do we find the exact values of \( a \) and \( b \) from a quadratic equation?
5. What are some common methods to solve quadratic equations?

**Tip:** Always check if a quadratic equation can be factored easily before using the quadratic formula or discriminant analysis.

Learn how to solve the quadratic equation 2x² - 14x + 3k - 4 = 0 given the condition a² - b² = 21. Understand the discriminant and its role in determining the value of k. Suitable for high school students studying algebra and quadratic equations.

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