To determine the value(s) of $b$ that will cause the equation $27x^2 + bx + 3 = 0$ to have one real solution, we use the discriminant of a quadratic equation.

For a quadratic equation of the form $ax^2 + bx + c = 0$, the discriminant $\Delta$ is given by:

$\Delta = b^2 - 4ac$

If the equation has one real solution, the discriminant must be zero:

$b^2 - 4ac = 0$

In this case:

• $a = 27$
• $b = b$ (unknown, what we are solving for)
• $c = 3$

Substituting these values into the discriminant formula:

$b^2 - 4(27)(3) = 0$

Simplify:

$b^2 - 324 = 0$

Solving for $b$:

$b^2 = 324$ $b = \pm \sqrt{324}$ $b = \pm 18$

Thus, the values of $b$ that will cause the equation to have one real solution are 18 and -18.

The correct answer is therefore:

• 18, -18

Would you like further details or have any questions?

Here are some follow-up questions for further practice:

1. What is the discriminant, and how does it determine the nature of the solutions of a quadratic equation?
2. If $b = 21$ in the equation $27x^2 + bx + 3 = 0$, how many solutions would there be and what type?
3. How does changing the constant term $c$ affect the values of $b$ that result in one real solution?
4. What would the value of $c$ need to be if $b = 18$ for the equation to still have one real solution?
5. If the quadratic equation had $a = 1$ instead of $a = 27$, what values of $b$ would make the discriminant zero?

Tip: Remember, when the discriminant is positive, the quadratic equation has two real solutions; if it's zero, there is one real solution; and if it's negative, there are no real solutions (only complex).