Question

How do you solve $ \left( x-4 \right)\left( x-3 \right)=0 $ ?

Answer 1

Hint: We have a product of two factors. We will check that the product of the factors is a quadratic equation. We will use the fact that the product of two numbers can be zero only if one of the two numbers is zero. Using this fact we will equate both the factors to zero and find the possible values of the variable.

Complete step by step answer:
The given equation is $ \left( x-4 \right)\left( x-3 \right)=0 $ . Let us look at the product of the given factors. When we multiply them, we get the following expression,
$\Rightarrow$ $ {{x}^{2}}-4x-3x+12=0 $
Simplifying this equation, we get
$\Rightarrow$ $ {{x}^{2}}-7x+12=0 $
We can see that we have obtained a quadratic equation. Now we will solve this quadratic equation. We already know the factors of the above quadratic equation. So, we can directly use the factorization method. The factorization of the above equation is the same as the given equation, which is
$\Rightarrow$ $ \left( x-4 \right)\left( x-3 \right)=0 $
We know that the product of two factors is zero only if one of the factors is zero. Hence, we have either $ x-4=0 $ or $ x-3=0 $ . Therefore, either $ x=4 $ or $ x=3 $ . The solution of the given equation is $ x=4 $ and $ x=3 $ .

Note:
 In the factorization method, we find the factors of the given quadratic equation. Since we already made the quadratic equation by multiplying the factors, we do not have to go through the process of factorization for solving the equation. If we had to solve the quadratic equation method using the completing square or the quadratic formula method, then we would have to go through the entire process of the said method. $