Quadratic Equations - GMAT Math Study Guide
A quadratic equation takes the form ax2 + bx + c = 0
- Quadratic Equation - An equation that can be written in the form ax2 + bx + c = 0.
For example, 2x2 + 3x + 2 = 0 is a quadratic equation while 3x + 2 is not a quadratic equation.
- Factoring - The process of breaking apart of an equation into factors (or separate terms) such that when the separate terms are multiplied together, they produce the original equation.
For example, x2 - x - 2 = (x+1)(x-2). In this case, the equation x2 - x - 2 = 0 can be broken apart into two factors [i.e., (x+1)(x-2) = 0] such that when these two separate terms (i.e., factors) are multiplied together, the result is the original equation.
- Root - A solution to an equation such as a quadratic equation such that f(x) = 0.
For example, for the quadratic equation x2 + x - 12 = 0, x = 3 and x = -4 are both roots since f(3) = 32 + 3 - 12 = 0 and f(-4) = (-4)2 - 4 - 12 = 0
Examples of Simple Quadratic Equations
Although all quadratic equations by definition fit the form ax2 + bx + c = 0, the most common simple format for a quadratic equation is as follows:
x2 + 6x + 9 = 0
(a = 1, b = 6, c = 9)
x2 - 4x + 4 = 0
(a = 1, b = -4, c = 4)
x2 + 2x - 35 = 0
(a = 1, b = 2, c = -35)
Examples of Complex or Untraditional Quadratic Equations
While most quadratic equations do appear in the format that the above examples appeared in, this is by no means the only format for a quadratic equation. The following are also quadratic equations:
2x2 - 6x + 10 = 0
(a = 2, b = -6, c = 10)
5x2 - 20 = 0
(a = 5, b = 0, c = -20)
9x2 = 81
(a = 9, b = 0, c = -81 since we could subtract 81 from both sides)
Solving Quadratic Equations
There are two main ways of solving a quadratic formula. The first method, the quadratic formula, works regardless of what format the quadratic equation comes in. The second method, factoring, becomes much more difficult as the quadratic equation becomes more complex. For example, it is much easier to factor a quadratic equation in the form ax2 + bx + c where a = 1 than it is to factor a quadratic equation in the form ax2 + bx + c where a ≠ 1.
Solving Quadratic Equations: The Quadratic Formula
While the quickest means to solve a quadratic equation is often through factoring (discussed next), a quadratic equation can be too complex to easily factor or it simply does not factor. In these instances, the quadratic formula can be used as it will always find the correct answer--regardless of the properties of the equation.
For a quadratic equation, which has the form ax2 + bx + c = 0, the roots are given by the formula.:
The ± within the quadratic formula indicates that x takes two values. A quadratic equation will have two solutions, known as roots, although these solutions are not necessarily unique. Consider the following examples:
Another means of solving a quadratic equation is through factoring. The essence of factoring, as defined above, is the process of breaking apart of an equation into factors (or separate terms) such that when the separate terms are multiplied together, they produce the original equation. Factoring works on the following fundamental relationship:
(x - r1)(x - r2) = 0 = x2 + bx + c
where r1 and r2 are the roots, or solutions, of the quadratic equation
Roots: -5, +2
(x+5)(x - 2) = ?
(x+5)(x - 2) = x2 -2x + 5x - 10
x2 -2x + 5x - 10 = x2 + 3x - 10
The Steps of Factoring
In order to learn the steps of factoring, we will use the example from above that we solved using the quadratic equation.
x2 + 2x - 8 = 0
1.) Find two numbers whose sum is b (or 2 in this example) and whose product is c (or -8 in this example).
Two such numbers are -2 and + 4, which add to +2 and multiply to -8.
x2 + 2x - 8 = 0
(x - 2)(x + 4) = 0
x = +2, -4 since these two numbers make each factor (i.e., [x - 2] or [x + 4]) equal zero.
Types of GMAT Problems
- Solve by Factoring
Some quadratic equations can be easily solved with the use of factoring. At times, a quadratic equation may be hiding with some terms of the quadratic on each side of the equals sign. In order to factor, move all of the terms to one side so that the squared term has a positive sign.
x2 = 9x - 20; Solve for x.
Correct Answer: C
- Begin by moving all the terms to the left side of the equation.
x2 - 9x + 20 = 0
- Factor the quadratic equation (which is now in quadratic form).
x2 - 9x + 20 = 0
You need two numbers that add to -9 and multiply to +20. These numbers are -5 and -4
x2 - 9x + 20 = (x - 5)(x - 4) = 0
- Now solve for x.
(x - 5)(x - 4) = 0
Either (x - 5) = 0 or (x - 4) = 0
If (x - 5) = 0, then x = 5
If (x - 4) = 0, then x = 4
- x = 4 or x = 5
- Solve by Using the Quadratic Formula
Some quadratic equations cannot be factored and require the use of the quadratic formula. Although a problem that absolutely requires the quadratic formula is extremely rare on the GMAT, it is nonetheless helpful to know how to use the quadratic formula.
Given the expression x2 + 6x + 6 = 0, solve for x
Correct Answer: A
- Solve by Using Substitution
Sometimes it may be easiest to solve the problem by substituting the answers into the equation and seeing which one satisfies the equation.
15x2 - 60x - 180 = 0; x = ?
Correct Answer: E
- Divide all the terms by 15 in order to simplify the problem:
15x2 - 60x - 180 = 0
x2 - 4x - 12 = 0
- There are two ways to do this problem: (1) Guess and check by substituting the answers into the question. (2) Algebra.
Guess and Check.
- In guess and check, you take each answer and plug it into the question equation and see if the answer makes the question equation true. If the answer makes the equation in the question true, then the answer is correct. If the answer does not fit into the equation in the question, then you try another answer. You continue this iterative process until you find the correct answer.
- Try Answer A: x = 5 -> (5)2 - 4(5) - 12 = -7 -> wrong
- Try Answer B: x = -3 -> (-3)2 - 4(-3) - 12 = 9 -> wrong
- Try Answer C: x = 3 -> (3)2 - 4(3) - 12 = -15 -> wrong
- Try Answer D: x = 2 -> (2)2 - 4(2) - 12 = -16 -> wrong
- Try Answer E: x = -2 -> (-2)2 - 4(-2) - 12 = 0 -> correct
- Note: You should also check the other solution to answer E in order to be thorough. Since you have ruled out A-D, you know E must be right. However, to be sure E is the correct answer, you should check the other x value since both must be true in order for E to be truly correct.
- Try Answer E: x = 6 -> (6)2 - 4(6) - 12 = 0 -> correct
- In factoring, we need a pair of numbers that multiply to -12 and add to -4. The factors of twelve are 1, 2, 3, 4, 6, 12. As a result, if we choose -6 and +2, we have two numbers that add to -4 and multiply to -12.
- x2 - 4x - 12 = 0
(x-6)(x+2) = 0
x = 6, -2