Question

All the time, in algebra, when it asks us to solve for the equation why do we always have to set our equation equal to zero? For example: x^2-4x=-4. To solve this, we have to set it equal to zero x^2-4x+4=0 which is (x-2) ^2=0 —> x=2. Why?

Answer 1

In algebra, setting an equation equal to zero is a common technique used to solve for the unknown variable. The reason for this is that when we set an equation equal to zero, we can often use factoring or the quadratic formula to solve for the variable.

In the example you gave, x^2-4x=-4, we can set the equation equal to zero by adding 4 to both sides of the equation, which gives us x^2-4x+4=0. Notice that the left side of this equation can be factored into (x-2)^2, so we have (x-2)^2=0.

At this point, we can use the zero product property, which states that if the product of two or more factors is zero, then at least one of the factors must be zero. In this case, the only factor is (x-2)^2, which means that (x-2)^2=0 if and only if x-2=0. Solving for x gives us x=2, which is the solution to the original equation x^2-4x=-4.

So, setting an equation equal to zero allows us to use factoring and the zero product property to solve for the unknown variable. It is a useful technique that is often used in algebra, especially when solving quadratic equations.

Answer 2

Explanation:

Setting an equation equal to zero is a technique used in algebra to help us solve equations. When we set an equation equal to zero, we are essentially looking for the points on the graph where the function crosses the x-axis. In other words, we are looking for the roots or solutions of the equation, which are the values of x that make the equation true.

In your example, x^2-4x=-4, we want to solve for x. By adding 4 to both sides, we get x^2-4x+4=0. This is equivalent to (x-2)^2=0, which can be simplified to x-2=0. Therefore, x=2 is the solution to the equation.

Setting the equation equal to zero allows us to use factoring or the quadratic formula to solve for x, which can be more straightforward than trying to solve the original equation.