If the product of two consecutive positive odd numbers is 143, form the equation and find the two numbers using Sridhar Acharya's formula (the quadratic formula).

Q.If the product of two consecutive positive odd numbers is 143, form the equation and find the two numbers using Sridhar Acharya's formula (the quadratic formula).

Let the first odd number be \(x\), Then the next consecutive odd number is \((x + 2)\) According to the question, \(x(x + 2) = 143\) ⇒ \(x^2 + 2x - 143 = 0\) Comparing this with the standard quadratic form \(ax^2 + bx + c = 0\), we get: \(a = 1\), \(b = 2\), \(c = -143\) Using Sridhar Acharya's formula (the quadratic formula): \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) ⇒ \(x = \frac{-2 \pm \sqrt{2^2 - 4 × 1 × (-143)}}{2 × 1}\) ⇒ \(x = \frac{-2 \pm \sqrt{4 + 572}}{2}\) ⇒ \(x = \frac{-2 \pm \sqrt{576}}{2}\) ⇒ \(x = \frac{-2 \pm 24}{2}\) ⇒ \(x = -1 \pm 12\) Since the number must be positive, \(x = -1 + 12 = 11\) ∴ The other number is \(11 + 2 = 13\) ∴ The two consecutive positive odd numbers are 11 and 13.