Step 1: Manipulate the equation to match the preceding template. In this
example, you would need to set the equation equal to zero.
x^2 + 7x + 5 = −7
+ 7 = + 7
x^2 + 7x + 12 = 0Step 2: Rewrite the equation in factored form: x^2 + 7x + 12 = (x + __ )(x + __).Step 3: Determine the values for the slots. To get the factors of the equation,
you need to find two integer values that add to yield your b and that multiply
to yield your c. In the preceding equation, 7 is your b and 12 is your c. What
two values multiply to 12 and add to 7? 3 and 4.
In the slots, you will put the two integer values that you arrived at in
Step 2. Thus in its factored form, the equation is (x + 3)(x + 4) = 0.Step 4: Solve for x. To solve for x, you must recognize an essential fact: any
time a product of two or more factors is zero, at least one of those factors must
have a value of zero.In the preceding example, if (x + 3)(x + 4) = 0, then either:
(x + 3) = 0 or (x + 4) = 0
x = −3 x = −4
So the roots of this equation are −3 and −4. Note that if you plug either of
these values into the original equation, you will arrive at a true statement.Setting the Quadratic Equation Equal to Zero
Oftentimes, you will be presented with a quadratic equation that does not appear to
match the preceding template.For example, if 4x^2 = x, then x =?
Seeing x on both sides of the equation, many students are tempted to divide
both sides of the equation by x to arrive at:
4 x = 1
x =^14
Though^14 is certainly a solution to the equation, the hypothetical student
committed an error here when dividing by x. Why? Because the student
essentially eliminated one of the solutions for x! Instead of arriving at two
solutions, the student arrived at only one.284 PART 4 ■ MATH REVIEW03-GRE-Test-2018_173-312.indd 284 12/05/17 11:55 am