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How do you solve quadratic equations by the new and improved Factoring AC Method?
Nghi1350
Solving quadratic equations by the new improved factoring AC Method.
This method aims to factor a given quadratic equation in standard form ax^2 + bx + c = 0 into 2 binomials in x by replacing the term bx by the 2 terms b1x and b2x that satisfy these 3 conditions: 1) the product b1*b2 = ac. 2) The sum (b1 + b2) = b. 3) Application of the Rule of Signs for Real Roots of a quadratic equation into the solving process.
Recall the Rule of Signs.
If a and c have different signs, the 2 real roots have different signs
Example: The equation 5x^2 - 4x - 9 = 0 have 2 roots with different signs.
If a and c have same sign, Both roots have same sign.
- If a and b have same sign, both roots are negative.
Example: The equation 5x^2 + 7x + 2 = 0 has 2 real roots both negative.
- If a and b have different signs, both real rooots are positive.
Example: The equation 7x^2 - 9x + 2 = 0 has 2 real roots both positive
When a = 1 - Solving equation type x^2 +bx + c = 0.
Solving results in finding 2 numbers knowing their sum -b and their product ac. In this case, solving by the improved Factoring AC Method is simple, fast and does't require neither factoring by grouping nor solving the 2 binomials.
Example 1. Solve: x^2 + 11x - 102 = 0. Solution. Roots have different signs. Compose factor pairs of ac = -102 with all first number of the pair being negative. Proceeding: (-1, 102)(-2, 51)(-3, 34)(-6, 17). Stop. This last sum is
-6 + 17) = 11 = b. Then, the sum of the 2 real roots is: -b = (6 - 17). The 2 real roots are 6 and -17. No factoring and solving binomials!
Example 2. Solve: x^2 + 31x + 108 = 0. Solution. Both real roots are negative. Compose factor pairs of ac = 108 with all numbers of the pairs being positive. Proceeding: (1, 108)(2, 48)(3, 32)(4, 24). Stop. This last sum is (4 + 24) = 28 = b. Then, the sum of the 2 real roots are: -b = (-4, -28). The 2 real roots are: -4 and -28.
When a not 1 - Solving equation type ax^2 + bx + c = 0.
The new method proceeds to find b1 and b2, then factor by grouping the equation into 2 binomials. Next, it solves the 2 binomials for x as usual. Remember these 2 TIPS when composing the factor pairs of ac.
TIP 1. When roots have different signs (a and c different signs), compose factor pairs of ac with all first number of the pair being negative.
TIP 2. When roots have same sign (a and c same sign), compose factor pairs of ac with all numbers positive.
Example 3. Solve: 8x^2 - 22x - 13 = 0. Solution. Roots have different signs. Compose factor pairs of ac = -104 with all first number of the pair being negative. Proceeding: (-1, 104)(-2, 52)(-4, 26). OK. This last sum is -4 + 26 = 22 = -b. Then, the sum (b1 + b2) = b = (4, -26). Next, replace in the equation the term (-22x) by the 2 terms (4x) and (-26x):
8x^2 + 4x - 26x - 13 = 4x(2x +1) - 13(2x + 1) = (2x + 1)(4x - 13) = 0
Next, solve the 2 binomials:
(2x + 1) = 0 --> x = -1/2
(4x - 13) = 0 --> x = 13/4
Example 4. Solve: 12x^2 + 29x + 15 = 0. Solution. Both real roots are negative. Compose factor pairs of ac = 180 with all positive numbers. Proceeding:(1, 180(2, 90)(3, 60)(4, 45)(5, 36)(9, 20). OK. This last sum is:
(9 + 20) = 29 = b. Then, b1 = 9 and b2 = 20.
Replace in the equation the term (29x) by the 2 terms (9x) and (20x) then factor the equation. Next, solve the 2 binomials:
(4x + 3) = 0 --> x = -3/4
(3x + 50 = 0 --> x = -5/3.
Conclusion. The new and improved Factoring AC Method improves the solving process in many ways.
a. When a = 1, It helps make the solving process simple, fast, systematic, with no factoring by grouping, and no solving binomials.
b. When a is not 1, it helps reduce the number of test cases in half.
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Is the foil method used for factoring quadratic equations?
Yes FOIL method can be used with quadratic expressions and equations
Can all quadratic equations be solved?
Well, that depends on what you mean "solve by factoring." For any quadratic equation, it is possible to factor the quadratic, and then the roots can be recovered from the factors. So in the very weak sense that every quadratic can be solved by a method that involves getting the factors and recovering the roots from them, all quadratic equations can be solved by factoring. However, in most cases, the only way of factoring the quadratic in the first place is to first find out what its roots are, and then use the roots to factor the quadratic (any quadratic polynomial can be factored as k(x - r)(x - s), where k is the leading coefficient of the polynomial and r and s are its two roots), in which case trying to recover the roots from the factors is redundant (since you had to know what the roots were to get the factors in the first place). So to really count as solving by factoring, it makes sense to require that the solution method obtains the factors by means that _don't_ require already knowing the roots of the polynomial. And in this sense, most quadratic equations are not solvable through factoring.
How many existing methods are there in solving quadratic equations?
There are 5 existing methods in solving quadratic equations. For the first 4 methods (quadratic formula, factoring, graphing, completing the square) you can easily find them in algebra books. I would like to explain here the new one, the Diagonal Sum Method, recently presented in book titled:"New methods for solving quadratic equations and inequalities" (Trafford 2009). It directly gives the 2 roots in the form of 2 fractions, without having to factor the equation. The innovative concept of the method is finding 2 fractions knowing their Sum (-b/a) and their Product (c/a). It is very fast, convenient and is applicable whenever the given quadratic equation is factorable. In general, it is hard to tell in advance if a given quadratic equation can be factored. However, if this new method fails to find the answer, then we can conclude that the equation can not be factored, and consequently, the quadratic formula must be used. This new method can replace the trial-and-error factoring method since it is faster, more convenient, with fewer permutations and fewer trials.
What is the third step in solving this equation by completing the square?
Completing the square is one method for solving a quadratic equation. A quadratic equation can also be solved by several methods including factoring, graphing, using the square roots or the quadratic formula. Completing the square will always work when solving quadratic equations and is a good tool to have. Solving a quadratic equation by completing the square is used in a lot of word problems.I want you to follow the related link that explains the concept of completing the square clearly and gives some examples. that video is from brightstorm.
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
Is the foil method used for factoring quadratic equations?
Yes FOIL method can be used with quadratic expressions and equations
When solving a quadratic equation by factoring what method is used?
Start with a quadratic equation in the form � � 2 � � � = 0 ax 2 +bx+c=0, where � a, � b, and � c are constants, and � a is not equal to zero ( � ≠ 0 a =0).
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.
Can all quadratic equations be solved?
Well, that depends on what you mean "solve by factoring." For any quadratic equation, it is possible to factor the quadratic, and then the roots can be recovered from the factors. So in the very weak sense that every quadratic can be solved by a method that involves getting the factors and recovering the roots from them, all quadratic equations can be solved by factoring. However, in most cases, the only way of factoring the quadratic in the first place is to first find out what its roots are, and then use the roots to factor the quadratic (any quadratic polynomial can be factored as k(x - r)(x - s), where k is the leading coefficient of the polynomial and r and s are its two roots), in which case trying to recover the roots from the factors is redundant (since you had to know what the roots were to get the factors in the first place). So to really count as solving by factoring, it makes sense to require that the solution method obtains the factors by means that _don't_ require already knowing the roots of the polynomial. And in this sense, most quadratic equations are not solvable through factoring.
Which the easy way the method of factoring or the solving the quadratic equation?
By knowing how to use the quadratic equation formula.
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)
Can all quadratics be solved by completing the square?
Completing the square is one method for solving a quadratic equation. A quadratic equation can also be solved by factoring, using the square roots or quadratic formula. Solving quadratic equations by completing the square will always work when solving quadratic equations-You can also use division or even simply take a GCF, set the quantities( ) equal to zero, and subtract or add to solve for the variable
What is the quadratic formula used for?
The quadratic formula is used all the time to solve quadratic equations, often when the factors are fractions or decimals but sometimes as the first choice of solving method. The quadratic formula is sometimes faster than completing the square or any other factoring methods. Quadratic formula find: -x-intercept -where the parabola cross the x-axis -roots -solutions
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.
How many existing methods are there in solving quadratic equations?
There are 5 existing methods in solving quadratic equations. For the first 4 methods (quadratic formula, factoring, graphing, completing the square) you can easily find them in algebra books. I would like to explain here the new one, the Diagonal Sum Method, recently presented in book titled:"New methods for solving quadratic equations and inequalities" (Trafford 2009). It directly gives the 2 roots in the form of 2 fractions, without having to factor the equation. The innovative concept of the method is finding 2 fractions knowing their Sum (-b/a) and their Product (c/a). It is very fast, convenient and is applicable whenever the given quadratic equation is factorable. In general, it is hard to tell in advance if a given quadratic equation can be factored. However, if this new method fails to find the answer, then we can conclude that the equation can not be factored, and consequently, the quadratic formula must be used. This new method can replace the trial-and-error factoring method since it is faster, more convenient, with fewer permutations and fewer trials.
Which part of the quadratic formula tells you whether the quadratic equation can be solved by factoring?
Here are two ways to know if a given quadratic equations can be factored (can be solved by factoring). 1. Calculate the Discriminant D = b^2 - 4ac. When D is a perfect square (its square root is a whole number), then the given equation can be factored. 2. Solve the equation by using the new Diagonal Sum method (Amazon e-book 2010). This method directly finds the 2 real roots without having to factor the equation. Solving usually requires fewer than 3 trials. If this method fails to get the answer, then we can conclude that the equation can not be factored, and consequently the quadratic formula must be used.
Which method of solving quadratic equations should be used when only an estimated solution is necessary?
Graphing
