- D If xa = 1, it is possible that a = 0 and that x equals any number (since any
number raised to the power of zero = 1). However, that possibility is eliminated
by the information that a > 0. So what can x be? It would appear that x must
equal 1, since 1 raised to any power equals 1. However, what if a is an even
integer? In that case, x can equal −1, since a negative raised to an even power
yields a positive result. If x = 1, the two quantities are equal. If x = −1, Quantity
B is greater. Thus the relationship cannot be determined. - C Since the quantities compare c to an expression that contains a and b, you
should manipulate the given equation to isolate c. First, cross-multiply:
b
√c =
a
1
→ ba = √c. Next, square both sides: ba2
= √c2
→ ba^22 = c. The quantities are equal.Quadratic Equations
So far, you have looked only at linear equations. In linear equations, there will
always be one solution for a given variable. In contrast, in quadratic equations, a
variable will usually have more than one solution. How do you know you have a
quadratic equation?
You have a quadratic equation whenever at least one of the variables in the
equation is raised to an even exponent.
Let’s say you are asked to evaluate (−4)^2. Using PEMDAS, you know that the
result is (−4) × (−4) = 16. Now let’s say that you are asked to evaluate 4^2. You get
4 × 4 = 16. Note that both 4 and −4 gave the same result. Why? Because raising a
variable to an even exponent will always produce a positive result.
Now let’s flip it:
If x^2 = 16, what are the possible values of x?
You may be tempted to calculate the square root of 16 and say that x = 4. But
watch out for the even exponent! Since the exponent on x is even, there will be a
positive and a negative solution for x. Thus the two solutions are x = 4 or −4.
Other forms of quadratic equations are:x^2 + 5x = −6 a^2 = a (^3) x = x
Common Templates of Quadratic Equations
On the GRE, quadratic equations will take a few common forms. The most
common form is
ax^2 + bx + c = 0
When presented with a quadratic equation in the preceding form, you will usually
be asked to find the solutions of that equation. To do so, you will need to factor.
Let’s look at an example:
If x^2 + 7x + 5 = −7, what are the possible values for x?
CHAPTER 11 ■ ALGEBRA 283
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