Solve 𝑥 squared minus four 𝑥 plus four equals zero by factorising.
Factorising is the opposite or inverse of expanding or multiplying out parentheses. In this case, as the only common factor of the equation is one, we are going to need two parentheses. The first term in both of the brackets or parentheses must be 𝑥 as 𝑥 multiplied by 𝑥 is equal to 𝑥 squared.
In order to work out the second term in both of the parentheses, we need to consider the general quadratic equation 𝑎 𝑥 squared plus 𝑏 𝑥 plus 𝑐 equals zero. If the coefficient of 𝑥 squared, 𝑎, is equal to one, then the two numbers have a product equal to 𝑐 and a sum equal to 𝑏. In our example, this means that the two numbers must have a product of four i.e., they have to multiply to give four. And they have to have a sum of negative four.
We have four possible pairs of numbers that multiply or have a product of four. Two multiplied by two equals four, negative two multiplied by negative two is also equal to four, four multiplied by one is equal to four, and negative four multiplied by negative one also gives us an answer of four.
Which of these four pairs of numbers gives us a sum of negative four when we add them? Well, two plus two is equal to positive four. So the top line cannot be our answer. Negative two plus negative two, however, does give us negative four. Carrying on, we can eliminate the bottom two lines as four plus one is equal to five. And negative four plus negative one is equal to negative five. This means that the numbers that are in our parentheses are negative two and another negative two. This leaves us with two brackets or parentheses 𝑥 minus two multiplied by 𝑥 minus two.
We could expand this fully to get back to our original equation. But in this case, we’ve been asked to solve it. As 𝑥 minus two multiplied by 𝑥 minus two is equal to zero, one of these individual parentheses must also be equal to zero: either the first bracket 𝑥 minus two is equal to zero or the second bracket 𝑥 minus two is equal to zero. Adding two to both sides of these equations to balance them gives us answers of 𝑥 equals two or 𝑥 equals two.
The majority of quadratic equations would have two different answers at this stage. However, since our brackets or parentheses were identical, we have ended up with just one solution: 𝑥 equals two. Therefore, the solution or answer to the equation 𝑥 squared minus four 𝑥 plus four equals zero is 𝑥 equals two. We could substitute this value back into the original equation to ensure we got an answer of zero.