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Solving quadratic equations by the new Transforming Method.
It proceeds through 3 Steps.
STEP 1. Transform the equation type ax^2 + bx + c = 0 (1) into the simplified type x^2 + bx + a*c = 0 (2), with a = 1, and with C = a*c.
STEP 2. Solve the transformed equation (2) by the Diagonal Sum Method that immediately obtains the 2 real roots y1, and y2.
STEP 3. Divide both y1, and y2 by the coefficient a to get the 2 real roots of the original equation (1): x1 = y1 /a, and x2 = y2/a.
Example 1. Solve: 12x^2 + 5x - 72 = 0 (1). Solve the transformed equation: x^2 + 5x - 864 = 0. Roots have different signs (Rule of Signs). Compose factor pairs of a*c = -864 with all first numbers being negative. Start composing from the middle of the factor chain to save time. Proceeding:.....(-18, 48)(-24, 36)(-32, 27). This last sum is -32 + 27 = -5 = -b. Then, the 2 real roots of (2) are: y1 = -32, and y2 = 27. Back to the original equation (1), the 2 real roots are: x1 = y1/a = -32/12 = -8/3, and x2 = y2/a = 27/12 = 9/4.
Example 2. Solve 24x^2 + 59x + 36 = 0 (1). Solve the transformed equation x^2 + 59x + 864 = 0 (2). Both roots are negative. Compose factor pairs of a*c = 864 with all negative numbers. To save time, start composing from the middle of the factor chain. Proceeding:....(-18, -48)(-24, -36)(-32, -27). This last sum is -59 = -b. Then 2 real roots of equation (2) are: y1 = -32 and y2 = -27. Back to the original equation (1), the 2 real roots are: x1 = y1/24 = -32/24 = -4/3, and x2 = y2/24 = -27/24 = -9/8.
To know how does this new method work, please read the article titled: "Solving quadratic equations by the new Transforming Method" in related links.
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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 are the pros and cons of the quadratic equation?
Pros: There are many real life situations in which the relationship between two variables is quadratic rather than linear. So to solve these situations quadratic equations are necessary. There is a simple equation to solve any quadratic equation. Cons: Pupils who are still studying basic mathematics will not be told how to solve quadratic equations in some circumstances - when the solutions lie in the Complex field.
How do you use the quadratic formula to solve a quadratic equation?
You convert the equation to the form: ax2 + bx + c = 0, replace the numeric values (a, b, c) in the quadratic formula, and calculate.
Why cant the zero product property be used to solve every quadratic equation?
If the discriminant of a quadratic equation is less than zero then it will not have any real roots.
Solve the quadratic equation y2-15 equals 0?
y=±√15
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
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)
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.
How do you solve a quadratic equation graphically?
The roots of the quadratic equation are the x-intercepts of the curve.
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.
How do you formulate quadratic equation that can be solve by extracting square roots?
By using the quadratic equation formula
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.
The quadratic formula cannot be used to solve an equation if the coefficient of the equation's x2-term is?
The quadratic formula cannot be used to solve an equation if the coefficient of the equation's x2-term is 0.
How do you solve a quadratic equation that will not factor?
How you solve an equation that doesn't factor is to plug a quadratic equation's format; ax2+bx+c into the quadratic formula which is x=-b+square root to (b2-4ac)/2a.
How many ways are there to solve a quadratic equation?
There are so far 8 common methods to solve quadratic equations:GraphingFactoring FOIL methodCompleting the square.Using the quadratic formula (derived from algebraic manipulation of "completing the square" method).The Diagonal Sum Method. It quickly and directly gives the 2 real roots in the form of 2 fractions. In fact, it can be considered as a shortcut of the factoring method. It uses the Rule of Signs for Real Roots in its solving process. When a= 1, it can give the 2 real roots quickly without factoring. Example. Solve x^2 - 39x + 108 = 0. The Rule of Signs indicates the 2 real roots are both positive. Write the factor-sets of c = 108. They are: (1, 108), (2, 54), (3, 36)...Stop! This sum is 36 + 3 = 39 = -b. The 2 real roots are 3 and 36. No needs for factoring! When a is not one, this new method selects all probable root-pairs, in the form of 2 fractions. Then it applies a very simple formula to see which root-pair is the answer. Usually, it requires less than 3 trials. If this new method fails, then this given quadratic equation can not be factored, and consequently the quadratic formula must be used. Please see book titled:"New methods for solving quadratic equations and inequalities" (Amazon e-book 2010).The Bluma MethodThe factoring AC Method (Youtube). This method is considerably improved by a "new and improved AC Method", recently introduced on Google or Yahoo Search.The new Transforming Method, recently introduced, that is may be the best and fastest method to solve quadratic equations. Its strong points are: simple, fast, systematic, no guessing, no factoring by grouping, and no solving the binomials. To know this new method, read the articles titled:"Solving quadratic equations by the new Transforming Method" on Google or Yahoo Search.BEST METHODS TO SOLVE QUADRATIC EQUATIONS. A. When the equation can't be factored, the best choice would be the quadratic formula. How to know if the equation can't be factored? There are 2 ways:1. Start solving by the new Transforming Method in composing factor pairs of a*c (or c). If you can't find the pair whose sum equals to (-b), or b, then the equation can't be factored.2. Calculate the Discriminant D = b^2 - 4ac. If D isn't a perfect square, then the equation can't be factored.B. When the equation can be factored, the new Transforming Method would be the best choice.
Why quadratic equation is used?
It's quite convenient, for it offers a general method to solve any equation that involves a polynomial of degree two (in one variable).
What is the quadratic formula cannot be used to solve an equation if the coefficient of the equation x square term is?
The quadratic formula cannot be used to solve an equation if the coefficient of the equation x square term is what?
