Question

How do you solve $ {{x}^{2}} $ − x − 1 = 0 using the quadratic formula?

Answer 1

Hint:
The quadratic formula states that the solutions to the quadratic equation $ a{{x}^{2}} $ + bx + c = 0 are directly obtained by using the formula: x = $ \dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a} $, where a ≠ 0. What are the values of a, b and c in the given equation?

Complete Step by step Solution:
Comparing the given quadratic equation $ {{x}^{2}} $ − x − 1 = 0 with the general form of the quadratic equations $ a{{x}^{2}} $ + bx + c = 0, we see that a = 1, b = − 1 and c = −1.
Using the formula x = $ \dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a} $ , we get:
x = $ \dfrac{-(-1)\pm \sqrt{{{(-1)}^{2}}-4(1)(-1)}}{2(1)} $
On multiplying and simplifying the terms further, we get:
⇒ x = $ \dfrac{1\pm \sqrt{1+4}}{2} $
⇒ x = $ \dfrac{1\pm \sqrt{5}}{2} $

⇒ x = $ \dfrac{1+\sqrt{5}}{2} $ OR x = $ \dfrac{1-\sqrt{5}}{2} $ , which are the two required solutions.

Note:
The solution to an expression of the form p × q × r = 0 is that at least one of p, q or r must be 0, i.e. p = 0 OR q = 0 OR r = 0.
In order to solve a quadratic equation, we complete two squares of the form $ {{a}^{2}} $ − $ {{b}^{2}} $ = 0 and then factorize as (a + b)(a − b) = 0 and get the solutions as a = −b OR a = b.