There is a rule that can help solve what would otherwise be difficult algebra problems on the SAT. You might find several places on your test where it’s applicable. The rule is:

**When two polynomial equations are set equal to each other, the **(boldsymbol x^{mathbf2})** coefficients MUST be equal to each other.**

This makes logical sense, as the (x^2) coefficient controls whether a parabola opens up or down, and its width. If equations are equal, it means their lines on the graph are exactly the same, so the** **(x^2) coefficients have to be the same.

Here is an example:

$$8x^2+15x-56=abx^2+10x-20x+20+25x-76$$

In the equation above, if *b* = 4, what does *a* equal?

**Strategy****:**

**Foil or distribute if one side is factored.****Identify which coefficients on the left side of the equals sign correspond to which coefficients on the right side. For example,**(mathbf5boldsymbol x^{mathbf2}boldsymbol=boldsymbol aboldsymbol x^{mathbf2})**, the****5 and the***a*correspond.**Write out mini equations of things that are equal. With**(mathbf5boldsymbol x^{mathbf2}boldsymbol=boldsymbol aboldsymbol x^{mathbf2})**you would write**(mathbf5boldsymbol=boldsymbol a)**.****Form systems of equations if necessary.**

Now back to our example:

We have quadratics set equal to each other. So, the (x^2) coefficients must be equal.

$$8x^2=abx^2$$

We also know that *b* = 4, so substituting gives us:

$$8x^2=4ax^2$$

In order for these to be equal, *a* must be 2.

Here’s a slightly less obvious usage of the rule.

$$4x^2-9=(px+t)(px-t)$$

In the equation above, *p* and *t* are constants. Which of the following could be the value of *p* ?

A) 2

B) 3

C) 4

D) 9

If we FOIL the right side, we’ll get

$$p^2x^2-t^2$$

By our rule, we know the (x^2) coefficients must be equal. So:

$$p^2=4$$

Taking the square root of both sides tells us *p* could be 2.

Login

Accessing this course requires a login. Please enter your credentials below!