4 Quadratics and polynomials

Lesson

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.

Solve \dfrac{x^{2}}{16}-3=6 for x.

Worked Solution

Idea summary

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** 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.

Solve (x-6)(x+7)=0 for x.

Worked Solution

Idea summary

**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.

Solve simple quadratic equations using a range of strategies.