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The standard form of a quadratic equation in one variable is ax^2 + bx + c = 0.
Beyond the 4 existing common solving methods, there is a new method, called the Diagonal Sum Method, presented in book titled:" New methods for solving quadratic equations and inequalities" (Amazon e-books 2010), that directly obtains the 2 real roots, in the form of 2 fractions, without factoring. It uses a Rule of signs for real roots and a Rule for the Diagonal Sum.
Example of using the Rule of signs for real roots.
The equation 15x^2 - 46x - 17 = has 2 real roots that have opposite signs.
The equation 21x^2 + 34x + 8 = 0 has 2 real roots, both negative.
The equation 21x^2 - 68x + 15 = 0 has 2 real roots, both positive.
The diagonal sum of a pair of 2 real roots.
Given a pair of two real roots: (c1/a1) and (c2/a1).
Their product is c1.c2/a1.a2 = c/a.
Their sum is (c1/a1) + (c2/a2) = (c1a2 + c2a1)/a1.a2 = -b/a
The sum (c1a2 + c2a1) is called the diagonal sum. The diagonal sum of the 2 real roots should be equal to -b.
Rule for the Diagonal Sum.
The diagonal sum of a true real root pair must be equal to (-b). If it is equal to b, the answers are opposite in sign. If a is negative, the above rule is reversal in sign.
A. When a = 1. Solving quadratic equation: x^2 + bx + c = 0.
When a =1, solving this quadratic equation by the new method is simple, fast, and doesn't requires factoring
Example 1. Solve: x^2 - 21x - 72 = 0.
Solution. Rule of signs shows the two roots have opposite signs.
Write factors-sets of c = -72: (-1, 72) (-2, 36) (-3, 24) Stop! This sum is 21 = -b.
The 2 real roots are: -3 and 24. No factoring!
Note. There are factors-sets in opposite sign (1, -72)(2, -36)...but they can be ignored since they give opposite diagonal sums. By convention, always put the negative sign in front of the first number.
Example 2. Solve: x^2 - 39x + 108 = 0.
Solution. Both roots are positive. Write factors-sets of c = 108.
They are: (1, 108) (2, 54) (3, 36)...Stop! This sum is 3 + 36 = 39 = -b.
The 2 real roots are: 3 and 36.
Example 3. Solve: x^2 + 27x + 50 = 0.
Solution. Both roots are negative. Factors-sets of c = 50: (-1, -50)(-2, -25)...Stop!. This sum is -27 = -b. The real roots are: -2 and -25.
B. When a and c are prime/small numbers.
The new method directly selects the probable root pairs from the (c/a) setup. The numerator of the setup contains all factor pairs of c. The denominator contains all factor pairs of a.
Example 4. Solve: 7x^2 + 90x - 13 = 0.
Solution. Roots have opposite signs. Write the c/a setup. The numerator contains unique factors pair of c = -13: (-1, 13). The denominator contains unique factor pair of a = 7 that is always kept positive: (1, 7). There is unique probable root pair: (-1/7 & 13/1). The other pair can be ignored since 1 is not a real root. The diagonal sum of the unique pair is -1 + 91 = 90 = b. According to the Rule for the diagonal sum, when the diagonal sum equals b, the real roots are: 1/7 and -13
Example 5. Solve: 7x^2 - 57x + 8 = 0.
Solution. Both roots are positive. Constant c = 8 has 2 factors pairs (1, 8), (2, 4). The c/a setup: (1, 8),(2, 4)/(1, 7) leads to 3 probable root pairs: (1/7 & 8/1) (2/1 & 4/7)(2/7 & 4/1). The diagonal sum of first set is:1 = 56 = 57 = -b. The real roots are 1/7 and 8.
Example 6. Solve: 6x^2 - 19x - 11 = 0.
Solution. Roots have opposite signs. Constant a = 6 has 2 factors-sets:(1, 6) (2, 3). The c/a setup: (1, 11)/(1, 6)(2, 3) give 3 probable roots pairs: (-1/6 & 11/1) (-1/2 & 11/3) (-1/3 & 11/2)
The second set has as diagonal sum: (22 - 3 = 19 = -b). The 2 real roots are: -1/2 and 11/3.
Note. There are opposite sign roots-pairs (1/6 & -11/1) (1/2 & -11/3)... but they can be ignored since they all give opposite diagonal sums.
C. When a and c are large numbers and contains themselves many factors.
These cases are considered complicated because the (c/a) setup contains many factor pairs in both numerator and denominator. In this case, the Diagonal Sum Method can transform a complicated multiple step solving process into a simplified one by doing some elimination operations. To know how to solve these complicated cases, please read the article "Solving complicated cases of quadratic equations" on this Wiki Answers website.
NOTE. The Diagonal Sum proceeds solving by basing on the c/a setup. That is why, this method may be called: The (c/a) Method.
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In mathematics how does one use the quadratic formula?
The Quadratic formula in mathematics is used to solve quadratic equations in algebra. The simplest way to solve these equations is to set each of the factors to zero and then solve each factor separately.
How do you solve quadratic equations by graphing?
josh hutcherson
What is the history of quadratic equations?
at first the first person to solve the quadratic equation is from the middle kingdom of Egypt. Greeks were also able to solve the quadratic equation but that was on the unproper way. Greeks were able to solve the quadratic equation by geometric method or equlid's method. equlid's method contains only three quadratic equation. dipohantus have also solved the quadratic equations but he have solved by giving only two roots any they both were only of positive signs.After that arbhatya also gave the two formulas for quadratic equation but the bentaguptahave only accepted only one of them after theat some of the Indian mathematican have also solved the quadratic equation who gave the proper definations and formula and in this way quadratic equation have been formed. Prabesh Regmi Kanjirowa National School
How do you simplify quadratic equations?
You can combine equivalent terms. You should strive to put the equation in the form ax2 + bx + c = 0. Once it is in this standard form, you can apply the quadratic formula, or some other method, to solve it.
How does the zero product rule help you solve quadratic equations?
Factor it! Set each equal to zero! Solve
In mathematics how does one use the quadratic formula?
The Quadratic formula in mathematics is used to solve quadratic equations in algebra. The simplest way to solve these equations is to set each of the factors to zero and then solve each factor separately.
How do you solve quadratic equations by graphing?
josh hutcherson
Can you solve a quadratic equation without factoring?
using the quadratic formula or the graphics calculator. Yes, you can do it another way, by using a new method, called Diagonal Sum Method, that can quickly and directly give the 2 roots, without having to factor the equation. This method is fast, convenient and is applicable to any quadratic equation in standard form ax^2 +bx + c = 0, whenever it can be factored. It requires fewer permutations than the factoring method does, especially when the constants a, b, and c are large numbers. If this method fails to get answer, then consequently, the quadratic formula must be used to solve the given equation. It is a trial-and-error method, same as the factoring method, that usually takes fewer than 3 trials to solve any quadratic equation. See book titled:" New methods for solving quadratic equations and inequalities" (Trafford Publishing 2009)
How did the Babylonians solve quadratic equations?
1+1=2
How do you solve simultaneous quadratic equations?
Graphically might be the simplest answer.
How do you solve quadratic equations by factoring?
There is a new method, called Diagonal Sum Method, that quickly and directly give the 2 roots without having to factor the equation. The innovative concept of this method is finding 2 fractions knowing their sum (-b/a) and their product (c/a). It is fast, convenient and is applicable to any quadratic equation in standard form ax^2 + bx + c = 0, whenever it can be factored. If it fails to find answer, then the equation is not factorable, and consequently, the quadratic formula must be used. So, I advise you to proceed solving any quadratic equation in 2 steps. First, find out if the equation can be factored? How?. Use this new method to solve it. It usually takes fewer than 3 trials. If its fails then use the quadratic formula to solve it in the second step. See book titled:" New methods for solving quadratic equations and inequalities" (Trafford Publishing 2009)
What does quadratic equations using the factoring method means?
It means you are required to "solve" a quadratic equation by factorising the quadratic equation into two binomial expressions. Solving means to find the value(s) of the variable for which the expression equals zero.
What is the quadratic formula for?
The quadratic formula is used to solve the quadratic equation. Many equations in which the variable is squared can be written as a quadratic equation, and then solved with the quadratic formula.
What is the history of quadratic equations?
at first the first person to solve the quadratic equation is from the middle kingdom of Egypt. Greeks were also able to solve the quadratic equation but that was on the unproper way. Greeks were able to solve the quadratic equation by geometric method or equlid's method. equlid's method contains only three quadratic equation. dipohantus have also solved the quadratic equations but he have solved by giving only two roots any they both were only of positive signs.After that arbhatya also gave the two formulas for quadratic equation but the bentaguptahave only accepted only one of them after theat some of the Indian mathematican have also solved the quadratic equation who gave the proper definations and formula and in this way quadratic equation have been formed. Prabesh Regmi Kanjirowa National School
What is the quadratic equation used for?
It is used to solve quadratic equations that cannot be factored. Usually you would factor a quadratic equation, identify the critical values and solve, but when you cannot factor you utilize the quadratic equation.
Why do you use square roots to solve quadratic equations?
Because it's part of the quadratic equation formula in finding the roots of a quadratic equation.
How do you simplify quadratic equations?
You can combine equivalent terms. You should strive to put the equation in the form ax2 + bx + c = 0. Once it is in this standard form, you can apply the quadratic formula, or some other method, to solve it.
