The Value of a/b b/a for Distinct Roots of x2 - 9x 5 0

Understanding the relationship between the coefficients of a quadratic equation and the properties of its roots is a fundamental concept in algebra. This article explores the value of a/b b/a for distinct roots of the quadratic equation x2 - 9x 5 0. We will derive the solution step by step, demonstrating the application of algebraic identities and factorization techniques.

Deriving the Roots

We start with the given quadratic equation:

x2 - 9x 5 0

The roots of a general quadratic equation ax2 bx c 0 can be found using the quadratic formula:

x (-b ± √(b2 - 4ac)) / 2a

For our equation, a 1, b -9, c 5. By substituting these values into the formula, we find the roots a and b to be:

x (9 ± √(81 - 20)) / 2 (9 ± √61) / 2

Thus, the roots are:

a (9 √61) / 2 and b (9 - √61) / 2

Sum and Product of Roots

The sum and product of the roots of the quadratic equation can be expressed as:

a b -b/a -(-9) 9

ab c/a 5

Deriving a/b b/a

We aim to find the value of a/b b/a. Let's start with the algebraic identity:

(a/b) (b/a) (a2b b2a) / (ab)

By squaring the sum of the roots:

(a b)2 a2 b2 2ab

Substituting the known values:

92 a2 b2 2(5)

81 a2 b2 10

a2 b2 71

Now, we can find:

(a/b) (b/a) (a2b b2a) / (ab) (a2b b2a) / 5

Since a2b b2a 71(ab) 71(5) 355

Thus, (a/b) (b/a) 355 / 5 71 / 5

Therefore, the value of a/b b/a is:

a/b b/a 71/5 or 14.2

Conclusion

For distinct roots of the quadratic equation x2 - 9x 5 0, the value of a/b b/a is 71/5 or 14.2. This value can be derived using basic algebraic identities and the relationships between the coefficients of the quadratic equation and its roots.

Understanding such algebraic manipulations can be valuable in solving complex problems in mathematics and related fields, demonstrating the powerful nature of algebraic techniques and identities.