If 2xsquare-(b-4)x-4(b+2)=0 has equal roots,find the possible values of b

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!

If 2x^2 - (b - 4)x - 4(b + 2) = 0 has equal roots, find the possible values of b.

Discover how to find the possible values of b for the quadratic equation 2x^2 - (b - 4)x - 4(b + 2) = 0, where the roots are equal. This solution involves using the discriminant condition and solving quadratic equations. Suitable for high school students in Grades 9-12.

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