Question

How do you solve \[{x^2} + 2x - 5 = 0\] ?

Answer 1

Hint: In this question, the equation has two distinct real roots, so we will solve this equation by using the quadratic formula. Convert the given expression into a quadratic equation and then factor it completely. We can find the factors of the given equation by the method of finding the roots of the quadratic equation.

Complete Step by Step Solution: -
We have been given with the equation in the question, \[{x^2} + 2x - 5 = 0\] , if we try to solve it by using the method of splitting the middle term, we won’t be able to do it as when -5 and coefficient of first term, i.e., 1 will be multiplied, we’ll get the answer -5 for which the factors are 1 and -5, which when added cannot give us the result as 2. Therefore, the method we will use to solve this equation is the quadratic formula.
So, first of all, we need to find the discriminant of the quadratic equation in the question. If the discriminant of quadratic equation is greater than 0 then, we get two distinct real roots, if it is less than 0, then there are no real roots and if the discriminant is equal to 0, then there will be two equal real roots.
Discriminant is calculated by formula –
$D = {b^2} - 4ac$
In the quadratic equation in the question, \[{x^2} + 2x - 5 = 0\] , we have –
$
  a = 1 \\
  b = 2 \\
  c = - 5 \\
 $
Putting these in the formula of discriminant –
$
   \Rightarrow D = {2^2} - 4 \times 1 \times \left( { - 5} \right) \\
   \Rightarrow D = 4 + 20 \\
   \Rightarrow D = 24 \\
 $
Hence, the discriminant of the equation is 24 which is greater than 0 so, there will be two distinct real roots.
Now, we know that, quadratic formula is –
$x = \dfrac{{ - b \pm \sqrt D }}{{2a}}$
We already got the values for $a,b$ and $D$. Therefore, putting all these values in the quadratic formula –
$ \Rightarrow x = \dfrac{{ - 2 \pm \sqrt {24} }}{{2 \times 1}} \cdots \left( 1 \right)$
Now, we have to find the square root of $24$ -
$
  \sqrt {24} = \sqrt {4 \times 6} \\
   \Rightarrow \sqrt {24} = 2\sqrt 6 \\
 $
Putting the value of $\sqrt {24} $ in equation (1) –
$ \Rightarrow x = \dfrac{{ - 2 \pm 2\sqrt 6 }}{{2 \times 1}}$
Further solving, we get –
$ \Rightarrow x = - 1 \pm \sqrt 6 $

Hence, the roots of the equation are $ - 1 + \sqrt 6 $ and $ - 1 - \sqrt 6 $.

Note:
The quadratic formula should be remembered by the student as it is an important formula to solve the questions of this type and using it, we can easily find the roots. The value of discriminant tells us the nature of roots before finding the value of roots which makes the equation easy to solve.