Question

How do you solve \[{x^2} = 100\]?

Answer 1

Hint: Here, we will solve the given quadratic equation for the variable. We will use the method of Prime Factorization to find the factors of the number. Then by using the factors we will find the square root of the number. On further solving it we will find the value of the variable. Prime Factorization is a method of finding the factors of a number in terms of prime numbers.

Complete step by step solution:
We are given with a Quadratic equation \[{x^2} = 100\]
Now, we will solve for the variable \[x\] by solving the given quadratic equation.
By taking square root on both the sides of the equation, we get
\[ \Rightarrow \sqrt {{x^2}} = \sqrt {100} \]
Since the square root and square are inverses to each other, they cancel each other. Thus, we get
\[ \Rightarrow x = \sqrt {100} \]
Now, we will find the square root of the number by using the method of prime factorization, we get
\[\begin{array}{l}2\left| \!{\underline {\,
  {100} \,}} \right. \\2\left| \!{\underline {\,
  {50} \,}} \right. \\5\left| \!{\underline {\,
  {25} \,}} \right. \\5\left| \!{\underline {\,
  5 \,}} \right. \\{\rm{ }}\underline 1 \end{array}\]
Thus, the factors of 100 is \[2 \times 2 \times 5 \times 5\].Thus, we get
\[ x = \sqrt {2 \times 2 \times 5 \times 5} \]
The factors \[2,5\] can be expressed in pairs, so by simplifying we get
\[ \Rightarrow x = \pm \left( {2 \times 5} \right)\]
By simplifying, we get
\[ \Rightarrow x = \pm 10\]
\[ \Rightarrow x = 10;x = - 10\]

Therefore, the solution for a quadratic equation \[{x^2} = 100\] is \[x = 10\] and \[x = - 10\].

Note:
We know that the given equation is a quadratic equation. A quadratic equation is an equation with the highest power 2. It is not necessary to have all the \[{x^2}\] term, \[x\] term, and the constant term for a quadratic equation but it is necessary to have the highest power as 2. We know that the number ending with 0, is a perfect square number. Whenever we are finding the square root of a number, then the solution can be either positive or negative value or both the positive and negative values.