The ratio of two roots of equation ax² +bx+c=0. a0 be 4:5, then value of b^2/ac is 

Given the ratio of the roots of the quadratic equation \( ax^2 + bx + c = 0 \) is \( 4:5 \), we need to find the value of \( \frac{b^2}{ac} \).

### Let the roots be \( r_1 = 4k \) and \( r_2 = 5k \).

According to Vieta's formulas for the quadratic equation:
1. The sum of the roots \( r_1 + r_2 = -\frac{b}{a} \),
2. The product of the roots \( r_1 \times r_2 = \frac{c}{a} \).

### Step 1: Sum of the Roots
The sum of the roots is:
\[
r_1 + r_2 = 4k + 5k = 9k
\]
By Vieta's formula:
\[
-\frac{b}{a} = 9k
\]
Thus,
\[
b = -9ak
\]

### Step 2: Product of the Roots
The product of the roots is:
\[
r_1 \times r_2 = 4k \times 5k = 20k^2
\]
By Vieta's formula:
\[
\frac{c}{a} = 20k^2
\]
Thus,
\[
c = 20k^2 a
\]

### Step 3: Finding \( \frac{b^2}{ac} \)
Now, we calculate \( \frac{b^2}{ac} \):
\[
\frac{b^2}{ac} = \frac{(-9ak)^2}{a \times 20k^2 a}
\]
Simplifying:
\[
\frac{b^2}{ac} = \frac{81a^2k^2}{20k^2a^2} = \frac{81}{20}
\]

### Final Answer:
The value of \( \frac{b^2}{ac} \) is \( \frac{81}{20} \).

Would you like further clarification or have other questions related to this problem? Here are 5 more questions that extend from this:

1. How can we derive the discriminant using the given root ratio?
2. What happens when the ratio of the roots is different?
3. Can you apply this ratio approach to higher degree polynomials?
4. How does the nature of the roots change with different coefficients?
5. Can we derive a general formula for any ratio of roots?

**Tip:** Simplifying complex expressions often requires organizing the relationships step by step. Break down the expressions carefully for better accuracy.

Discover how to find the value of b^2/ac when the roots ratio of a quadratic equation ax^2 + bx + c = 0 is given as 4:5. Follow a detailed step-by-step solution using Vieta's formulas and algebraic manipulation. Suitable for high school 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