Introduction
A one-variable second-order equation , also called a second-order equation in one variable or, more often, a quadratic equation , can be written in the following standard form:
ax ^{2} + bx + c = 0
where a, b , and c are constants, and x is the variable. (The constant c here does not stand for the speed of light.) Equations of this type can have two real-number solutions, one real-number solution, or no real-number solutions.
Some Examples
Any equation that can be converted into the preceding form is a quadratic equation. Alternative forms are
mx ^{2} + nx = p
qx ^{2} = rx + s
( x + t ) ( x + u ) = 0
where m, n, p, q, r, s, t , and u are constants. Here are some examples of quadratic equations:
x ^{2} + 2 x + 1 = 0
−3 x ^{2} − 4 x = 2
4 x ^{2} = −3 x + 5
( x + 4) ( x − 5) = 0
Get It Into Form
Some quadratic equations are easy to solve; others are difficult. The first step, no matter what scheme for solution is contemplated, is to get the equation either into standard form or into factored form.
The first equation above is already in standard form. It is ready for an attempt at solution, which, we will shortly see, is rather easy.
The second equation can be reduced to standard form by subtracting 2 from each side:
−3 x ^{2} − 4 x = 2
−3 x ^{2} − 4 x − 2 = 0
The third equation can be reduced to standard form by adding 3 x to each side and then subtracting 5 from each side:
4 x ^{2} = −3 x + 5
4 x ^{2} + 3 x = 5
4 x ^{2} + 3 x − 5 = 0
The fourth equation is in factored form. Scientists and engineers like this sort of equation because it can be solved without having to do any work. Look at it closely:
( x + 4) ( x − 5) = 0
The expression on the left-hand side of the equals sign is zero if either of the two factors is zero. If x = −4, then the equation becomes
(−4 + 4) (−4 − 5) = 0
0 × −9 = 0 (It works)
If x = 5, then the equation becomes
(5 + 4) (5 − 5) = 0
9 × 0 = 0 (It works again)
It is the height of simplicity to “guess” which values for the variable in a factored quadratic will work as solutions. Just take the additive inverses (negatives) of the constants in each factor.
There is one possible point of confusion that should be cleared up. Suppose that you run across a quadratic like this:
x ( x + 3) = 0
In this case, you might want to imagine it this way:
( x + 0) ( x + 3) = 0
and you will immediately see that the solutions are x = 0 or x = −3.
In case you forgot, at the beginning of this section it was mentioned that a quadratic equation may have only one real-number solution. Here is an example of the factored form of such an equation:
( x − 7) ( x − 7) = 0
Mathematicians might say something to the effect that, theoretically, this equation has two real-number solutions, and they are both 7. However, the physicist is content to say that the only real-number solution is 7.
The Quadratic Formula
Look again at the second and third equations mentioned a while ago:
−3 x ^{2} − 4 x = 2
4 x ^{2} = −3 x + 5
These were reduced to standard form, yielding these equivalents:
−3 x ^{2} − 4 x − 2 = 0
4 x ^{2} + 3 x − 5 = 0
You might stare at these equations for a long time before you get any ideas about how to factor them. You might never get a clue. Eventually, you might wonder why you are wasting your time. Fortunately, there is a formula you can use to solve quadratic equations in general. This formula uses “brute force” rather than the intuition that factoring often requires.
Consider the standard form of a one-variable second-order equation once again:
ax ^{2} + bx + c = 0
The solution(s) to this equation can be found using this formula:
x = [− b ± ( b ^{2} − 4 ac ) ^{1/2} ]/2 a
A couple of things need clarification here. First, the symbol ±. This is read “plus or minus” and is a way of compacting two mathematical expressions into one. It’s sort of a scientist’s equivalent of computer data compression. When the preceding “compressed equation” is “expanded out,” it becomes two distinct equations
x = [− b + ( b ^{2} − 4 ac ) ^{1/2} ]/2 a
x = [− b − ( b ^{2} − 4 ac ) ^{1/2} ]/2 a
The second item to be clarified involves the fractional exponent. This is not a typo. It literally means the ½ power, another way of expressing the square root. It’s convenient because it’s easier for some people to write than a radical sign. In general, the z th root of a number can be written as the 1/ z power. This is true not only for whole-number values of z but also for all possible values of z except zero.
Plugging In
Examine this equation once again:
−3 x ^{2} − 4 x − 2 = 0
Here, the coefficients are
a = −3
b = −4
c = −2
Plugging these numbers into the quadratic formula yields
x = {4 ± [(−4) ^{2} − (4 × −3 × −2)] ^{1/2} }/(2 × −3)
= 4 ± (16 − 24) ^{1/2} /−6
= 4 ± (−8) ^{1/2} /−6
We are confronted with the square root of −8 in the solution. This a “non-real” number.
Those “nonreal” Numbers
Mathematicians symbolize the square root of −1, called the unit imaginary number , by using the lowercase italic letter i . Scientists and engineers more often symbolize it using the letter j , and henceforth, that is what we will do.
Any imaginary number can be obtained by multiplying j by some real number q . The real number q is customarily written after j if q is positive or zero. If q happens to be a negative real number, then the absolute value of q is written after − j . Examples of imaginary numbers are j 3, − j 5, j 2.787, and − j π.
The set of imaginary numbers can be depicted along a number line, just as can the real numbers. In a sense, the real-number line and the imaginary-number line are identical twins. As is the case with human twins, these two number lines, although they look similar, are independent. The sets of imaginary and real numbers have one value, zero, in common. Thus
j 0 = 0
A complex number consists of the sum of some real number and some imaginary number. The general form for a complex number k is
k = p + jq
where p and q are real numbers.
Mathematicians, scientists, and engineers all denote the set of complex numbers by placing the real-number and imaginary-number lines at right angles to each other, intersecting at zero. The result is a rectangular coordinate plane (Fig. 1-5). Every point on this plane corresponds to a unique complex number; every complex number corresponds to a unique point on the plane.
Fig. 1-5 . The complex numbers can be depicted graphically as points on a plane, defined by two number lines at right angles.
x = 4 ± j 8 ^{1/2} /−6
Now that you know a little about complex numbers, you might want to examine the preceding solution and simplify it. Remember that it contains (−8) ^{1/2} . An engineer or physicist would write this as j 8 ^{1/2} , so the solution to the quadratic is
Back To “reality”
Now look again at this equation:
4 x ^{2} + 3 x − 5 = 0
Here, the coefficients are
a = 4
b = 3
c = −5
Plugging these numbers into the quadratic formula yields
x = {−3 ± [3 ^{2} − (4 × 4 × −5)] ^{1/2} }/(2 × 4)
= −3 ± (9 + 80) ^{1/2} /8
= −3 ± (89) ^{1/2} /8
The square root of 89 is a real number but a messy one. When expressed in decimal form, it is nonrepeating and nonterminating. It can be approximated but never written out precisely. To four significant digits, its value is 9.434. Thus
x ≈ −6 ± 9.434/8
If you want to work this solution out to obtain two plain numbers without any addition, subtraction, or division operations in it, go ahead. However, it’s more important that you understand the process by which this solution is obtained. If you are confused on this issue, you’re better off reviewing the last several sections again and not bothering with arithmetic that any calculator can do for you mindlessly.
Practice problems for these concepts can be found at: Equations, Formulas, And Vectors for Physics Practice Test
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From Physics Demystified: A Self-Teaching Guide. Copyright © 2002 by The McGraw-Hill Companies, Inc. All Rights Reserved.