Solving Quadratic Equations
Solving quadratic equations involves finding the values of the variable that satisfy a second-degree polynomial equation, typically expressed in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. This mathematical concept has a rich history, with origins tracing back over 4,000 years to Babylonian mathematicians who tackled problems related to area and perimeter. Various methods have been developed to solve quadratic equations, including factoring, taking square roots, completing the square, and using the quadratic formula.
Factoring relies on the principle of zero products, while the square root method utilizes the properties of square roots to derive solutions. Completing the square transforms the equation into a perfect square trinomial, making it easier to solve. The quadratic formula, which is derived from completing the square, provides a straightforward arithmetic approach to find solutions for any quadratic equation. Additionally, the discriminant—a value derived from the coefficients—helps determine the nature of the solutions, indicating whether they are real or complex. Overall, mastering these methods is essential for students and anyone looking to understand the foundations of algebra.
Subject Terms
Solving Quadratic Equations
Quadratic equations are second degree polynomial equations, and they have the following standard form.
where is the variable, and are constants, and . Quadratic equations have been studied more than 4,000 years, dating back to Babylonian mathematicians who found the dimensions and of a rectangle with a given area and perimeter . See Figure 1.
In modern notation, the problem that the Babylonians formulated and solved is the following: Given and , find and such that
Combining and yields , which is a quadratic equation. The Babylonian solution was verbal, as opposed to formulaic. Thus, in English it would look something like the following.
Divide the perimeter by 4.
Square the result of step 1.
Subtract the area from the result of step 2.
Find the square root of the result of step 3.
Add the result of step 4 to that of step 1 to obtain the base.
In modern notation the above translates as , which is one of the two solutions that would be obtained using modern methods. Once is solved for, can be solved for using or . This ancient problem is still a standard textbook exercise for algebra students today.
In 628 AD, Brahmagupta published the general solution to the quadratic equation
which he gave verbally as, "To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value." This translates into modern notation as
which is one of the two solutions that would be obtained using modern methods. Finally, it was in 1637 that the French mathematician René Descartes published the solution of the quadratic equation in its modern form:
Factoring
The method of solving quadratic equations by factoring rests on the principle of zero products, which states that if and are real numbers, then:
Another way to phrase this is that it is impossible to multiply two nonzero real numbers together and get a product of zero. The same principle applies when and are real-valued expressions, and this fact is exploited to solve quadratic equations by factoring.
This same basic program is used whenever solving quadratic equations by factoring. In this example the equation has integer solutions. This is not always the case, as in the following example, which has rational solutions.
The expression is not always factorable using the methods of elementary algebra. In such cases equation is best solved using other methods, as discussed below.
Taking the Square Root
This method for solving quadratic equations is based on the principle of square roots, which states that if is a real number and is a non-negative real number, then:
The same principle applies when is a real-valued expression, and this can be used to solve quadratic equations by taking square roots.
The presence of the radical indicates that the solutions of the above equation are irrational. The factoring methods of elementary algebra are not able to turn up such solutions.
The presence of the imaginary unit indicates that the solutions of the above equation are complex. Again, elementary factoring methods fail to reveal such solutions.
Completing the Square
The method of completing the square is based on an observation made on perfect square trinomials:
In both and it is observed that if one takes half of the first degree coefficient and squares the result, one obtains the constant term of each polynomial . This provides a procedure for obtaining a perfect square trinomial. For instance, in the expression , the first degree coefficient is . Based on the above observation, adding the constant to the expression will result in a perfect square trinomial: . The addition of to completes the perfect square trinomial .
The method of solving quadratic equations by completing the square is based on the above in conjunction with the principle of square roots. It must be kept in mind that this method only works when the second degree coefficient is .
The Quadratic Formula
The quadratic formula is based on the method of completing the square. This is explicitly shown by using that method to solve , which is the most general quadratic equation. This is done below.
The quadratic formula reduces the solution of any quadratic equation to an arithmetic problem, as shown below.
The quantity that appears under the radical in is called the discriminant, and its value determines the nature of the solutions of .
If , then has two distinct real solutions.
Example: , , Solutions: .
If , then has a single real solution.
Example: , , Solution: .
If , then has two non-real complex conjugate solutions.
Example: , , Solutions: .
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