To find the possible values of $b$ such that the quadratic equation:

$2x^2 - (b - 4)x - 4(b + 2) = 0$

has equal roots, we need to use the discriminant condition for a quadratic equation.

For any quadratic equation of the form:

$ax^2 + bx + c = 0$

the discriminant is given by:

$\Delta = b^2 - 4ac$

For the equation to have equal roots, the discriminant must be zero, i.e., $\Delta = 0$.

Step 1: Identify $a$, $b$, and $c$

From the given equation:

$2x^2 - (b - 4)x - 4(b + 2) = 0$

We compare this to the standard quadratic form:

• $a = 2$
• $b = -(b - 4) = -b + 4$
• $c = -4(b + 2) = -4b - 8$

Step 2: Set up the discriminant equation

The discriminant for this equation is:

$\Delta = (b^2 - 4ac)$

Substituting the values for $a$, $b$, and $c$:

$\Delta = ( (-b + 4)^2 ) - 4(2)(-4b - 8)$

Step 3: Simplify the discriminant equation

First, simplify the square term:

$(-b + 4)^2 = b^2 - 8b + 16$

Now expand the second part:

$4(2)(-4b - 8) = 8(4b + 8) = 32b + 64$

Now, write the full discriminant equation:

$\Delta = (b^2 - 8b + 16) - (32b + 64)$

Simplify further:

$\Delta = b^2 - 8b + 16 - 32b - 64$

$\Delta = b^2 - 40b - 48$

Step 4: Solve for $\Delta = 0$

For equal roots, set the discriminant to zero:

$b^2 - 40b - 48 = 0$

This is a quadratic equation in $b$. Solve it using the quadratic formula:

$b = \frac{-(-40) \pm \sqrt{(-40)^2 - 4(1)(-48)}}{2(1)}$

Simplify:

$b = \frac{40 \pm \sqrt{1600 + 192}}{2}$ $b = \frac{40 \pm \sqrt{1792}}{2}$ $b = \frac{40 \pm 42.33}{2}$

Step 5: Find the two possible values for $b$

Now, calculate the two possible values of $b$:

1. $b = \frac{40 + 42.33}{2} = \frac{82.33}{2} = 41.165$
2. $b = \frac{40 - 42.33}{2} = \frac{-2.33}{2} = -1.165$

Thus, the possible values of $b$ are:

$b \approx 41.165 \quad \text{or} \quad b \approx -1.165$

Summary:

The two possible values of $b$ that satisfy the condition for equal roots are approximately:

$b \approx 41.165 \quad \text{or} \quad b \approx -1.165$

Do you need more details or have any questions?

Here are 5 related questions:

1. What is the significance of the discriminant in a quadratic equation?
2. How can we verify if roots are real, equal, or complex using the discriminant?
3. What are the next steps to factor the quadratic equation once $b$ is known?
4. How does changing the value of $b$ affect the shape of the parabola?
5. Can there be a quadratic equation with no real roots for certain values of $b$?

Tip: Always check your discriminant calculations carefully when solving for equal roots!