What value(s) of b will cause 27x2 + bx + 3 = 0 to have one real solution? Select all that apply.
A.	b = −18
B.	b = −9
C.	b = 9
D.	b = 18

To find the value(s) of \( b \) that will cause the quadratic equation \( 27x^2 + bx + 3 = 0 \) to have exactly one real solution, we need to check when the discriminant of the quadratic equation is equal to zero.

The discriminant \(\Delta\) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:

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

For the quadratic equation to have exactly one real solution, the discriminant must be zero:

\[
\Delta = 0
\]

Substituting the values from the equation \( 27x^2 + bx + 3 = 0 \) (where \( a = 27 \), \( b = b \), and \( c = 3 \)) into the discriminant formula:

\[
b^2 - 4(27)(3) = 0
\]

Simplify this:

\[
b^2 - 324 = 0
\]

To solve for \( b \):

\[
b^2 = 324
\]

\[
b = \pm \sqrt{324}
\]

\[
b = \pm 18
\]

So, the values of \( b \) that will cause the quadratic equation to have exactly one real solution are \( b = 18 \) and \( b = -18 \).

**Correct choices:**
- A. \( b = -18 \)
- D. \( b = 18 \)

Would you like more details or have any questions? Here are some additional related problems to consider:

1. What is the general form of a quadratic equation?
2. How does the discriminant determine the nature of the roots of a quadratic equation?
3. How many real solutions exist if the discriminant is positive?
4. How does completing the square relate to solving quadratic equations?
5. What are the conditions for a quadratic equation to have two distinct real solutions?

**Tip:** Remember, when solving quadratic equations, the discriminant is key to determining the nature of the solutions!

Discover how to determine values of b that result in one real solution for the quadratic equation 27x^2 + bx + 3 = 0. Explore the discriminant method and solve step-by-step. Suitable for advanced high school students.

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