Why do you have to set the LHS of a quadratic equation equal to zero to solve it?

Question

For example $5=x^2-4x+9$ , set this equation to zero by subtracting $5$ from each side. Why do we have to set it equal to zero to work it out?

Answer 1

So that we could be able to factorize it and use the $AB = 0$ Principle.

The $AB = 0$ is like this, if you have two numbers with products equal to zero, then either both numbers are zero, or one of then must be zero.

Consider your equation, $x^2 - 4x + 9 = 5$ now if we can set the the $LHS$ equal to zero, then we can factorize and use the $AB = 0$ principle.

As $x^2 - 4x + 4 = 0$ then $x(x - 2) -2(x - 2) = 0$ which yields $( x - 2)(x - 2) = 0$

For equality to hold, $(x - 2)$ must be equal to $0$, which is a clever way to find the roots of a quadratic equation.

Answer 2

You don't always have to set anything to zero. Consider the equation $x^2-4x = 12,$ which can be solved by the method of "completing the square" as follows: \begin{align} x^2-4x &= 12 \\ (x^2-4x+4) - 4 &= 12 \\ (x - 2)^2 - 4 &= 12 \\ (x - 2)^2 - 4 + 4 &= 12 + 4 \\ (x - 2)^2 &= 16 \\ x - 2 &= \pm 4 \\ \end{align} Hence either $x = 6$ or $x = -2.$

On the other hand, I can think immediately of two methods in which the first step is to gather everything on one side of the equation, leaving zero on the other side. One of these is the factorization method (already explained in another answer); the other is the usual quadratic formula, $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, $$ which assumes that you have put the equation into the form $ ax^2 + bx + c = 0.$ (How would you use this formula to solve $5=x^2-4x+9$ without subtracting $5$ from each side?)