Question

How do you use a quadratic formula to solve ${x^2} - 5 = 0$?

Answer 1

Hint: According to the question we have to determine the solution of the given quadratic expression ${x^2} - 5 = 0$ which is as mentioned in the question with the help of quadratic formula. So, first of all to determine the solution of the given quadratic with the help of quadratic formula we have to obtain the values of a, b, and c with the help of the general form of the quadratic expression which is as mentioned below:

Formula used: $ a{x^2} + bx + c = 0...............(A)$
Where, a is the coefficient of${x^2}$, b is the coefficient of x and c is the constant term.
Now, we have to substitute all the values in the formula which is the quadrant formula as mentioned below:
$ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}....................(B)$
Now, we have to solve the expression obtained after placing all the values of a, b and c which can be done by adding, subtracting, multiplying and dividing the terms of the expression.

Complete step-by-step solution:
Step 1: First of all to determine the solution of the given quadratic with the help of quadratic formula (A) we have to obtain the values of a, b, and c with the help of the general form of the quadratic expression which is as mentioned in the solution hint. Hence,
$ \Rightarrow a = 1,b = 0$ and
$ \Rightarrow c = - 5$
Step 2: Now, we have to substitute all the values in the formula which is the quadrant formula (B) as mentioned in the solution hint. Hence,
$ \Rightarrow x = \dfrac{{ - 0 \pm \sqrt {{0^2} - 4 \times 1 \times ( - 5)} }}{{2 \times 1}}$
Step 3: Now, we have to solve the expression obtained after placing all the values of a, b and c which can be done by adding, subtracting, multiplying and dividing the terms of the expression. Hence,
$ \Rightarrow x = \dfrac{{\sqrt {20} }}{2}$
Now, we have to find the square roots of the numerator of the fraction as obtained just above,
$ \Rightarrow x = \dfrac{{2\sqrt 5 }}{2}$
On eliminating 2 from the numerator and the denominator of the expression as obtained just above,
$ \Rightarrow x = \pm \sqrt 5 $

Hence, with the help of the formula (A) and (B) we have determined the required solution of the expression with the quadrant formula which is $x = \pm \sqrt 5 $.

Note: It is necessary that we have to obtain the values of a, b, and c which can be determined with the help of the general form of the quadratic expression which is $a{x^2} + bx + c = 0$ .
On solving a quadratic expression only two possible roots/zeroes can be determined which will satisfy the given quadratic expression means on substituting the values in the expression it will become 0.