Question

How do you solve \[{x^2} - 5x = 36\]?

Answer 1

Hint: Here, we will use the concept of factorization. First, we will write the given equation in the standard form of the quadratic equation. Then we will split the middle term of the equation and form factors of the equation by taking the common terms in the equation. Then we will apply zero product property to get the value of \[x\].

Complete step by step solution:
The given equation is \[{x^2} - 5x = 36\].
Factorization is the process in which a number is written in the form of its factors which on multiplication give the original number.
First, we will rewrite the given equation in the standard form of the quadratic equation.
Subtracting 36 from both sides of the equation, we get
\[ \Rightarrow {x^2} - 5x - 36 = 0\]
Now we will split the middle term into two parts such that their product will be equal to the product of the first term and the third term of the equation. Therefore, we get
\[ \Rightarrow {x^2} - 9x + 4x - 36 = 0\]
Now we will be taking \[x\] common from the first two terms and taking 4 commons from the last two terms. Therefore the equation becomes
\[ \Rightarrow x\left( {x - 9} \right) + 4\left( {x - 9} \right) = 0\]
Factoring out \[\left( {x - 9} \right)\] from the equation, we get
\[ \Rightarrow \left( {x - 9} \right)\left( {x + 4} \right) = 0\]
Now applying zero product property, we get
\[ \Rightarrow \left( {x - 9} \right) = 0\] or \[\left( {x + 4} \right) = 0\]
Simplifying the equation, we get
\[ \Rightarrow x = 9\] or \[x = - 4\]

Hence, by solving the equation \[{x^2} - 5x = 36\] we will get the value of \[x\] as \[x = 9\] and \[x = - 4\].

Note:
We can say that the given equation is a quadratic equation because the highest degree of the equation is 2. A quadratic equation has two solutions which we have founded by factoring the equation. Factors are the smallest part of the number or equation which on multiplication will give us the actual number of equations. We have also used zero product property, which states that when a product of two terms is equal to zero, then either of the terms is equal to zero.