If x² - 4x + 4b = 0 has two real solutions, find the value of 'b'.
A)b = 0 B)b < 1
C)b = +1, -1 D)b ≥ 1   slve

To determine the value of \( b \) for which the quadratic equation \( x^2 - 4x + 4b = 0 \) has two real solutions, we can use the **discriminant** method. The discriminant \( \Delta \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:

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

For the equation to have real solutions, the discriminant must be non-negative, i.e., \( \Delta \geq 0 \).

Given the equation:

\[
x^2 - 4x + 4b = 0
\]

Here, \( a = 1 \), \( b = -4 \), and \( c = 4b \).

Now, calculate the discriminant:

\[
\Delta = (-4)^2 - 4(1)(4b) = 16 - 16b
\]

For the equation to have real solutions, we need:

\[
\Delta \geq 0 \implies 16 - 16b \geq 0
\]

\[
16 \geq 16b \implies b \leq 1
\]

Thus, for the quadratic equation to have two real solutions, \( b \) must satisfy \( b \leq 1 \).

Therefore, the correct answer is:

**Option D) \( b \geq 1 \) does not hold.**  
**The correct range is \( b \leq 1 \).**  
But since it was phrased in reverse, correct understanding would imply the only answer covering \( b = 1 \) and \( b < 1 \) is **B)**.

### Do you want further details or have any questions?

Here are 5 related questions:
1. What are the different conditions on the discriminant for a quadratic equation to have distinct real, repeated, or complex solutions?
2. How does the vertex form of a quadratic equation relate to its discriminant?
3. Can a quadratic equation have exactly one real solution? Under what circumstances?
4. What happens to the roots of the quadratic equation when \( b = 1 \) in this problem?
5. How would the graph of \( x^2 - 4x + 4b = 0 \) change as \( b \) varies?

**Tip:** Always remember that the discriminant determines the nature of the roots of a quadratic equation.

Discover how to find the value of 'b' in the quadratic equation x^2 - 4x + 4b = 0 to ensure two real solutions. This problem explores the discriminant method and its application to quadratic equations. Suitable for high school students in Grades 10-12.

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