Once you have factorised a quadratic equation, you will have an expression that looks something like (x-A)(x-B)=0 where a and b are numbers. The solutions to this equation are then A and S (since putting x=A in the equation above makes the first bracket (A-A) which is zero, and therefore A is a solution. Check the same for x=B!)
Example Solve the quadratic equation x^2-2x-8=0 by factorising.
When we factorise we get: (x-4)(x+2)=0 For equation to hold, we must have either x=4 or x=-2, and so the solutions are 4 and -2.
Solving by the factoring AC Method has been so far the most popular one to solve quadratic equations in standard form ax^2 + bx + c = 0. This method aims to factor the quadratic equation into 2 binomials by finding 2 terms, b1* and b2* that satisfy these 2 conditions:1) the product b1*b2 = a*c and 2) The sum (b1 + b2) = b.
A. When a = 1 - Equation type x^2 + bx + c = 0.
Example1. Solve x^2 - 9x - 102 = 0. Find 2 numbers b1 and b2 that b1*b2 = -102, and (b1 + b2) = -9. Proceed by composing factor pairs of a*c = -102.. Proceeding:(-1, 102)(1, -102)(-2, 51)(2, -51)9-3, 34)(3, -34)(-6, 17)(6, -17). OK. Then b1 = 6 and b2 = -17. Next, replace in the equation the term -9x by the 2 terms (6x) and (-17x). We get: x^2 + 6x - 17x - 102 = x*(x - 17) + 6*(x - 17) = (x - 17)(x + 6). Then the 2 real roots are: 17 and -6.
B. When a is not 1 - Equation type ax^2 + bx + c = 0.
Example 2. Solve: 5x^2 + 6x - 8 = 0. Find b1 and b2 that b1*b2 = 6, and (b1 + b2) = -8. Proceed by composing factor pairs of a*c = -40. Proceeding:(-1, 40)(1, -40)(-2, 20)(2, -20)(-4, 10). OK. Then b1 = -4 and b2 = 10. Continue factoring by grouping:
5x^2 + 10x - 4x - 8 = 5x(x + 2) - 4(x + 2) = (x + 2)(5x - 4) = 0. Next solve the 2 binomials: (x + 2) = 0 --> x = -2; and (5x - 4) = 0 --> x = 4/5.
NOTE. There is a "new and improved factoring AC Method", recently introduced by Nghi H Nguyen on Google Search (or Yahoo), that considerably improves the AC Method.
When a = 1, the improved method can immediately obtain the 2 real roots without factoring by grouping and solving the binomials.
When a is not 1, this new method shows a better way to compose the factor pairs of a*c. To know about this new method, read the articles:"Solving quadratic equations by the new and improved factoring AC Method" on Yahoo or Google Search.
There is a new "Transforming Method", recently introduced by Nghi H Nguyen on Google or Yahoo Search, that may replace the factoring AC Method as the simplest and fastest method to solve quadratic equations in standard form ax^2 + bx + c = 0 that can be factored. The new method uses 3 features in its solving process:
1. The rule of Signs for Real Roots of a quadratic equation.
2. TheDiagonal Sum Method to solve equation type x^2 + bx + c = 0 (a = 1).
3. The transformation of a quadratic equation in standard form into a simplifed one with a = 1.
This Transforming Method proceeds in 3 steps.
STtEP 1. Transform the given quadartic eqution ax^2 + bx + c = 0 (1) into the simplified one with a = 1 and with C = a*c:. The transformed equation has the form: x^2 + bx + a*c = 0. (2).
STEP 2. Solve the transformed equation (2) by the Diagonal Sum Method that immediately obtains the 2 real roots y1 and y2 without factoring by grouping and solving binomials.
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. Solve: 16x^2 - 62x + 21 = 0.(1) Transformed equation: x^2 - 62x + 336 = 0. Both roots are positive (Rule of Signs). Compose factor pairs of a*c = 336. with all positive numbers. Proceeding: (1, 336)(2, 168)(4, 82)(6, 56). This last sum is 62 = -b. The 2 real roots of the transformed equation are: y1 = 56 and y2 = 6. Back to the original equation (1), the 2 real roots are: x1 = y1/16 = 56/16 = 7/2, and x2 = y2/16 = 6/16 = 3/8.
We see that the strong points of this new "Transforming Method" are: fast, simple, no guessing, systemetic, no factoring by grouping and no solving the binomials.
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