Find the values ​​of m so that the roots of the equation: (m-4)x²-4(m-5)x + 9 = 0 for both root positive

To find the values of \( m \) that make both roots of the quadratic equation

\[
(m-4)x^2 - 4(m-5)x + 9 = 0
\]

positive, we need to analyze the quadratic equation and apply specific conditions on the roots.

### Step 1: General form of the quadratic equation

The equation is already in the form:

\[
a x^2 + b x + c = 0
\]

where:
- \( a = m - 4 \)
- \( b = -4(m - 5) \)
- \( c = 9 \)

### Step 2: Conditions for both roots to be positive

For both roots to be positive, the following conditions must be satisfied:

1. **The discriminant must be non-negative** for real roots to exist:
   \[
   \Delta = b^2 - 4ac \geq 0
   \]
2. **Both roots must be positive**. This is achieved when:
   - The sum of the roots \( \frac{-b}{a} > 0 \)
   - The product of the roots \( \frac{c}{a} > 0 \)

### Step 3: Calculate the discriminant

The discriminant \( \Delta \) is given by:

\[
\Delta = b^2 - 4ac = \left[ -4(m-5) \right]^2 - 4(m-4)(9)
\]

Simplifying this:

\[
\Delta = 16(m-5)^2 - 36(m-4)
\]

Expand both terms:

\[
\Delta = 16(m^2 - 10m + 25) - 36(m - 4)
\]

\[
\Delta = 16m^2 - 160m + 400 - 36m + 144
\]

\[
\Delta = 16m^2 - 196m + 544
\]

For real roots to exist, we need:

\[
\Delta \geq 0
\]

Thus:

\[
16m^2 - 196m + 544 \geq 0
\]

### Step 4: Conditions on the sum of the roots

The sum of the roots is given by:

\[
\text{Sum of roots} = \frac{-b}{a} = \frac{-(-4(m-5))}{m-4} = \frac{4(m-5)}{m-4}
\]

For the sum of the roots to be positive, we require:

\[
\frac{4(m-5)}{m-4} > 0
\]

This inequality holds when both the numerator and the denominator have the same sign. So, \( m-5 \) and \( m-4 \) must either both be positive or both be negative. Therefore, the critical points are \( m = 4 \) and \( m = 5 \), and we must check intervals around these points.

### Step 5: Conditions on the product of the roots

The product of the roots is:

\[
\text{Product of roots} = \frac{c}{a} = \frac{9}{m-4}
\]

For the product of the roots to be positive, we require:

\[
\frac{9}{m-4} > 0
\]

This inequality holds when \( m-4 > 0 \), meaning \( m > 4 \).

### Step 6: Combine all conditions

- From the sum of the roots condition: \( m > 5 \)
- From the product of the roots condition: \( m > 4 \)

Thus, combining these, we get \( m > 5 \).

### Conclusion:

The values of \( m \) such that both roots of the quadratic equation are positive are:

\[
m > 5
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How do we calculate the roots of a quadratic equation?
2. What does the discriminant tell us about the nature of the roots?
3. How can the conditions on the sum and product of roots ensure positivity?
4. What would happen if \( m = 5 \)?
5. How can we graphically represent quadratic equations?

**Tip**: When solving quadratic equations, always check both the discriminant and conditions on the roots to ensure they meet specific criteria like positivity.

Find the values ​​of m so that the roots of the equation: (m-4)x²-4(m-5)x + 9 = 0 are both positive.

Solve for the values of m in the quadratic equation (m-4)x² - 4(m-5)x + 9 = 0 where both roots are positive. This solution covers algebraic concepts like the quadratic equation formula, discriminant calculation, and inequalities to ensure positive roots. Ideal for Grades 9-11 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation