What is the least positive real value of 'm' for which the equation 4x2 + 2(m-10)x4m-33 has real roots?

We are tasked with finding the least positive value of \( m \) such that the quadratic equation:

\[
4x^2 + 2(m-10)x + 4m - 33 = 0
\]

has real roots. For any quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant \( \Delta \) is given by:

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

Real roots exist if and only if \( \Delta \geq 0 \). Let's apply this to the given quadratic equation.

### Step 1: Identify coefficients

For the quadratic equation \( 4x^2 + 2(m-10)x + (4m - 33) = 0 \), we have:
- \( a = 4 \) (coefficient of \( x^2 \)),
- \( b = 2(m - 10) \) (coefficient of \( x \)),
- \( c = 4m - 33 \) (constant term).

### Step 2: Compute the discriminant

The discriminant \( \Delta \) for this equation is:

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

Substitute \( b = 2(m - 10) \), \( a = 4 \), and \( c = 4m - 33 \):

\[
\Delta = [2(m - 10)]^2 - 4(4)(4m - 33)
\]

Simplifying both terms:

\[
\Delta = 4(m - 10)^2 - 16(4m - 33)
\]

Expand both terms:

\[
\Delta = 4(m^2 - 20m + 100) - 16(4m - 33)
\]
\[
\Delta = 4m^2 - 80m + 400 - 64m + 528
\]
\[
\Delta = 4m^2 - 144m + 928
\]

### Step 3: Solve for \( \Delta \geq 0 \)

We now solve the inequality:

\[
4m^2 - 144m + 928 \geq 0
\]

To simplify, divide the entire inequality by 4:

\[
m^2 - 36m + 232 \geq 0
\]

Now, we solve the corresponding quadratic equation \( m^2 - 36m + 232 = 0 \) using the quadratic formula:

\[
m = \frac{-(-36) \pm \sqrt{(-36)^2 - 4(1)(232)}}{2(1)}
\]
\[
m = \frac{36 \pm \sqrt{1296 - 928}}{2}
\]
\[
m = \frac{36 \pm \sqrt{368}}{2}
\]
\[
m = \frac{36 \pm 19.2}{2}
\]

Thus, the two solutions are:

\[
m_1 = \frac{36 + 19.2}{2} = \frac{55.2}{2} = 27.6
\]
\[
m_2 = \frac{36 - 19.2}{2} = \frac{16.8}{2} = 8.4
\]

### Step 4: Determine the least positive value of \( m \)

Since we need the least positive value of \( m \), the answer is:

\[
m = 8.4
\]

Thus, the least positive real value of \( m \) for which the equation has real roots is \( \boxed{8.4} \).

---

Would you like more details or have any further questions?

Here are 5 questions to consider:
1. How do you use the discriminant to determine the nature of the roots of a quadratic equation?
2. What are the conditions for a quadratic equation to have real roots?
3. How is the quadratic formula applied to find the roots of a quadratic equation?
4. How do you handle inequalities involving quadratic expressions?
5. Can you explain how simplifying the discriminant equation leads to the solution?

**Tip:** Always check the discriminant when determining the nature of roots in a quadratic equation!

What is the least positive real value of 'm' for which the equation 4x² + 2(m-10)x + 4m-33 has real roots?

Discover how to find the least positive value of 'm' that makes the equation 4x² + 2(m-10)x + 4m-33 have real roots. This detailed solution involves applying the discriminant theorem and quadratic formula. Suitable for students in Grades 10-12.

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