Algebra Continued

Words of “Wiz-dom”-Now that we’ve covered the basics, let’s move on to some more sophisticated, although, not necessarily more complicated algebra.

Quadratic Formula-Infrequently the ACT requires you to solve a quadratic equation that can’t be factored so you’ll need to use the quadratic formula:

For example, for the equation:

Be careful with this formula. It is easy to forget it is “-b” and that there are both positive and negative signs in front of the radical. Keep in mind that most of the time you will come out with two solutions.

Completing the Square-In addition to the quadratic formula, you can also “complete the square” when confronted by an “unfactorable” quadratic equation. Some students find this an easier technique than using the quadratic formula. Considering the equation for the quadratic equation discussion above:  , use the steps described below:

More words of “Wiz-dom” on Completing the Square

First, move the constant to the other side of the equals sign (remember to change the constant’s sign),  becomes .

Second, take 1/2 the coefficient of x (6 in this case), square it (which gets rid of any negative values), and add it to both sides of the equation, 6/2 is 3, 3 squared is 9, and adding it to both sides gives us:

Third, factor the left side of the equation:

Fourth, extract the square root of both sides:

Words of “Wiz-dom”-When doing the last step, remember the square root of a number has both a positive and negative value.

Finally, solve for x:

Words of “Wiz-dom”-If you have trouble finishing the math section, skip any question that requires you to use the quadratic equation and come back to it. It takes a lot of time to do one of these for the measly, paltry, piddling, trifling reward! You’ll probably find quicker, easier points later in the math section. So mark it with a “?” and come back to it if you’ve got time.

Equations and Absolute Values-Be careful here. Like quadratic equations, there are usually going to be two possible solutions. For example, for the equation , you’ve got to keep in mind that x-7 can be either +4 or -4. Therefore, x is either 11 or 3.

So the simple thing to do is eliminate the absolute value signs. Set up the equation twice: once for a positive value and once for a negative value. Solve each equation. Practice with the following:

Answer to first question

Answer to second question

Slope-intercept Equation

Words of “Wiz-dom”-Plan on needing to use the slope-intercept equation. It’s a test writer favorite. Its general form is y=mx+b. All you need to do is isolate y on one side of the equation and apply the values on the other side. m represents the slope and b indicates the y-intercept.

Slope of an Equation-Solve for y and the coefficient of x is the slope. Be sure to watch for whether the coefficient is positive or negative. For example in the equation:

-1/2 is the slope.

Y-intercept of an Equation-Solve for y and the constant (“b” in the slope-intercept equation of y=mx+b) is the intercept. For example in the equation above, 1.5 is the y-intercept. Keep in mind that the graph of a line intercepts the y-axis when x is 0. Therefore, another way to do it is to solve for y and set x=0. The solution for y when x is 0 is then the y-intercept.

Sample Questions

1.

If x>0 and ?
A. 
B. 
C. 
D. 
E. 

Answer to question #1

2.

If  and x is a real number that is not prime, then x=
A. –3
B. 
C. 1
D. 
E. 3

Answer to question #2

3.

In the (x,y) coordinate plane, what are the slope and y-intercept of the line:
A. slope = -2 and intercept = -3
B. slope = -2 and intercept = +3
C. slope = +2 and intercept = -3
D. slope = +2 and intercept = +3
E. slope = +2 and intercept = -6

Answer to question #3

4.

What are the (x,y) coordinates of the unique point on the graph of  when ?
A. -5,3
B. -5,0
C. 2,-5
D. 3,-5
E. 3,5

Answer to question #4

5.

In the (x,y) coordinate plane, which of the following pairs of graphs form a right angle?
A
B. 
C. 
D. 
E. 

Answer to question #5

6.

Which of the following is the solution set for 8 being less than four less than one half of three times the variable n?
A. n<8
B. n<5
C. n>5
D. n>6
E. n > 8

Answer to question #6

7.

In the standard (x,y) coordinate plane, how many points do the following pair of graphs have in common?
A. 0
B. 1
C. 2
D. 3
E. infinitely many

Answer to question #7

8.

For the equation ,the slope and the y-intercept are
A. 1 and 0
B. 
C. 1 and 1
D. 
E. 

Answer to question #8

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