4.01 Quadratic equations
Quadratic equations
We call equations like x-7=2 linear equations. These are equations where the variable of interest has a power of 1. Linear equations have only one solution. That is, there is only one value of the variable which will make the equation true. In this case, the only solution is x=9.
If instead, the variable of interest in the equation has a power of 2, we call them quadratic equations. Quadratic equations can potentially have two solutions. For example, in the equation x^{2}-7=2, there are two solutions, x=3 and x=-3. We can also write this as x= \pm 3.
The symbol \pm means "plus or minus". We can use this as a shorthand for both the positive and negative of a number. We can check these solutions by substituting them into the original equation and seeing if it holds true.
Examples
Example 1
Solve \dfrac{x^{2}}{16}-3=6 for x.
Equations where the variable of interest has a highest power of one are linear equations. These have at most one solution.
Equations where the variable of interest has a highest power of two are quadratic equations. These have at most two solutions.
The symbol \pm means "plus or minus".
If we can rearrange a quadratic equation into the form x^{2}=k, then we can solve the equation by taking the positive and negative square roots. That is, x=\pm\sqrt{k}.
The null factor law
The null factor law states that if a product of two or more factors is equal to 0, then at least one of those factors must be equal to 0. For example, if xy=0 then either x=0 or y=0. We can use this to solve quadratic equations.
If we can rearrange a quadratic equation into the form (x-a)(x-b)=0 then we know that either x-a=0 or x-b=0. We can solve the quadratic equation by solving each of these linear equations.
Examples
Example 2
Solve (x-6)(x+7)=0 for x.
The null factor law states that if a product of two or more factors is equal to 0, then at least one of those factors must be equal to 0.
If we can rearrange a quadratic equation into the form (x-a)(x-b)=0 then we know that either x-a=0 or x-b=0. We can solve the quadratic equation by solving each of these linear equations.