Equations

Algebra: Equations

Read the following SAT test question and then select the correct answer. 

You should read every math problem on the SAT carefully.  Identify your bottom line, assess your options for reaching it, and then select the most efficient method to attack the problem.  Once you have an answer, loop back to make sure that it matches your bottom line.

y = x² - 4x + c

In the quadratic equation above, c is a constant. The graph of the equation in the xy-plane contains the points (-2, 0) and (6,0). What is the value of c?

Bottom Line: c = ?

Assess your Options:  For this problem, you are given two points (x, y).  That means that you could plug in the x and y values for either point and solve for c (this is the method that most students will use):

(-2, 0)	(6, 0)
0 = (-2)² - 4(-2) + c	0 = (6)² - 4(6) + c
0 = 4 – (-8) + c	0 = 36 – 24 + c
0 = 12 + c	0 = 12 + c
-12 = c	-12 = c

Isolating the variable in either of these two equations will get you the correct value of c.  However, notice how many steps there are.  Can you just look at the two points and know the answer?  Yes!  Think about you find the roots of an equation.

Attack the Problem:  Your original equation is already set equal to zero.  You know this because both of the points have a y value of 0.  In order to factor a polynomial, you need two binomials.  Here you already know that x = -2 or 6.  That means that your two binomials are (x + 2) and (x – 6).  Now c is the last number that you would get in your polynomial if you multiplied (x + 2) by (x – 6).  What number is that? -12!  

All you had to do was multiply the last two numbers (2 × -6) because every other combination would have an x.  If you don’t see how that works, multiply out (x + 2)(x – 6):

Use FOIL (First, Inner, Outer, Last)

x² + 2x – 6x – 12 (combine like terms)

x² - 4x – 12

When you compare this equation to the original equation, you will see that in place of the c you now have a -12.

Loop Back:  During a test, you would never work through a problem three times (time waster!), so this is where you would check to make sure that you solved for the correct variable.  Look down at your answer choices.

(A) -12

(B) -6

(C) 4

(D) 6

(E) 12

The correct answer is (A).

On sat.collegeboard.org, 51% of the responses were correct.

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