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A quadratic equation consists of a single variable with three terms in the standard form: ax2 + bx + c = 0. The first quadratic equations were developed as a method used by Babylonian mathematicians around 2000 BC to solve simultaneous equations. Quadratic equations can be applied to problems in physics involving parabolic motion, path, shape, and stability. Several methods have evolved to simplify the solution of such equations for the variable x. Any number of quadratic equation solvers, in which the values of the quadratic equation coefficients can be entered and automatically calculated, can be found online.
The three methods most commonly used to solve quadratic equations are factoring, completing the square, and the quadratic formula. Factoring is the simplest form of solving a quadratic equation. When the quadratic equation is in its standard form, it is easy to visualize if the constants a, b, and c are such that the equation represents a perfect square. First, the standard form must be divided through by a. Then, half of, what is now, the b/a term must be equal to twice, what is now, the c/a term; if this is true, then the standard form can be factored into the perfect square of (x ± d)2.
If the solution of a quadratic equation is not a perfect square and the equation cannot be factored in its present form, then a second solution method — completing the square — can be used. After dividing through by the a term, the b/a term is divided by two, squared, and then added to both sides of the equation. The square root of the perfect square can be equated to the square root of all the remaining constants on the right hand side of the equation in order to find x.
The final method of solving the standard quadratic equation is by directly substituting the constant coefficients (a, b, and c) into the quadratic formula: x = (-b±sqrt(b2-4ac))/2a, which was derived by the method of completing the squares in the generalized equation. The discriminant of the quadratic formula (b2 - 4ac) appears under a square root sign and, even before the equation is solved for x, can indicate the type and number of solutions found. The type of solution depends on whether the discriminant is equal to the square root of a positive or negative number. When the discriminant is zero, there is only one positive root. When the discriminant is positive, there are two positive roots, and when the discriminant is negative, there are both positive and negative roots.
The Diagonal Sum Method for solving quadratic equations ax^2 + bx + c = 0.
Concept of the method: Directly find 2 real roots, in the form of 2 fractions, knowing their product (c/a) and sum (-b/a).
Rule of signs for real roots:
1. When a and c have different signs, roots have opposite signs.
2. When a and c have same sign, roots have same sign.
a. If a and b have different signs, both roots are positive (+)
b. If a and b have same sign, both roots are negative (-).
Rule of the diagonal sum of a root pair.
Given a pair of real roots: (c1/a1, c2/a2).
Their product is equal to (c/a), meaning: c1c2 = c and
: a1a2 = a. The numerators of a root pair are a factor-pair of c. The denominators are a factor-pair of a.
Their sum: (c1/a1 + c2/a2) = (c1a2 + c2a1)/a1a2 = -b/a. The sum (c1a2 + c2a1) is called the diagonal sum. From there comes the rule:
"The diagonal sum of a true root-pair must equal to (-b). If it equals to (-b), the solution is the negative of this pair."
1. When a = 1. Solving x^2 + bx + c = 0.
The diagonal sum reduces to the sum of the 2 real roots. Solving is very fast, and doesn't need factoring, due to the Rule of signs
Example 1. Solve x^2 + 75x + 216 = 0. Both roots are negative. Write factor pairs of c = 216. They are: (-1, -216),(-2, -108),(-3, -72)...This sum is -3 - 72 = -75 = -b. The 2 real roots are -3 and -72.
1. When a is not 1. The DS Method directly selects the probable root pairs from the quotient (c/a).
a. If a and c are both prime numbers: the number of probable root pair is limited to one, except when 1 (or -1) is a real root.
Example 2. Solve 11x^2 - 142x - 13 = 0. Roots have different signs. There is unique probable root pair: (-1/11, 13/1). Its DS is 143- 1 = 142 = -b. The 2 real roots are -1/11 and 13.
b. If a and c are small numbers and may have one (or 2) factors. Most of the factorable quadratic equations given in classes/books belong to this case. In this case, we can write down all of the probable root pairs, then find the one that has a DS equal to -b (or b). Number of trials is always fewer than 4.
Example 3. Solve 10x^2 + 31x + 13 = 0. Both roots are negative. The constant a = 10 has 2 factor pairs: (1, 10) and (2, 5). There are 3 probable root pairs:
(-1/10, -13/1),(-1/2, -13/5),(-1/5, -13/2). The DS of the second pair is: -5 - 26 = -31 = -b. The 2 real roots are -1/2 and -13/5.
Beyond the four existing methods (graphing, completing the square, factoring, formula) there is a new method called Diagonal Sum Method.
Its concept is direct, finding the two real roots in the form of two fractions knowing their sum (-b/a) and product (c/a). It is very fast when a = 1 and when the constants a, c are prime/small numbers. It is a trial and error method, same as the factoring one, but it reduces in half the number of permutations.
It saves the time used to solve the 2 binomials for x. It is considered as a shortcut of the factoring method.
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