Question
Question: Solve the quadratic equation \[\sqrt 5 {x^2} + x + \sqrt 5 = 0\]...
Solve the quadratic equation 5x2+x+5=0
Solution
If we have a polynomial of degree ‘n’ then we have ‘n’ number of roots or factors. A polynomial of degree two is called a quadratic polynomial and its zeros can be found using many methods like factorization, completing the square, graphs, quadratic formula, etc. The quadratic formula is used when we fail to find the factors of the equation. In the given question we have to solve the given quadratic equation using the quadratic formula. That is x=2a−b±b2−4ac.
Complete step-by-step solution:
Given, 5x2+x+5=0.
On comparing the given equation with the standard quadratic equation ax2+bx+c=0. We get, a=5, b=1 and c=5.
Substituting in the formula of standard quadratic, x=2a−b±b2−4ac.
⇒x=2(5)−(1)±(1)2−4(5)(5)
⇒x=2(5)1±1−4(5)
⇒x=251±1−4(5)
⇒x=251±1−20
⇒x=251±−19
We know that −1=i
⇒x=251±i19
Thus, we have two roots,
⇒x=251+i19 and ⇒x=251−i19. This is the required result.
Note: On the x-axis, the value of y is zero so the roots of an equation are the points on the x-axis that is the roots are simply the x-intercepts. We cannot solve this by simple factorization. That is by expanding the middle term into a sum of two terms, such that the product of two terms is equal to the product of ‘a’ and ‘c’, the sum of two terms is equal to ‘b’. We also know that the quadratic formula is also known as Sridhar’s formula.