# I need a detailed method of doing, and double checking Algebraic Quadratic equations?

What I lacked in school, over 30 years ago, was a backbone to ask the teacher what he meant, but his hurried

*learn this*style of teaching. What I need now, is something I have having trouble getting, even from any of the several books on the subject I have, that claim to make it simple: that is, I need both the methodology of doing the equations, and I need to know how and why they work, so I can work out when I have done something wrong.
active| newest | oldestHowever, that would be highly unethical, and maybe impractical, especially if you're renting.

I think the next best thing to having a captive

humanmath expert, would be some sort of captive robot. Making robots do boring work is not unethical, unless it takes away the job of a union member, but that'a another story. Here's the part where I give you the pitch for Octave.http://octave.sourceforge.net/

If you want the ability to solve, or create arbitrary quadratic equations, Octave can do this.

Want to solve x

^{2}- 6x + 8 = 0 ? Use the roots() function:Want to create a quadratic equation with real roots at 5 and -3/2 ?

In other words: 2x

^{2}- 7x - 15 = 0Many expensive graphing calculators have similar functionality. Of course the advantage Octave has is it is free, minus the effort of installing it, figuring out how to use it, etc.

#*1. The Rule of Signs for Real Roots of a quadratic equation.

#*2. The Diagonal Sum Method that solves and immediately obtains the 2 real roots of equation type: x^2 + bx + c = 0.

#*3. The transformation of quadratic equations type ax^2 + bx + c = 0 into the simplified form, with a = 1.

#*The NewTransforming Method. It proceeds through 3 Steps.

#*STEP 1. Transform the equation type ax^2 + bx + c = 0 (1) into the transformed equation form: x^2 + bx + a*c = 0 (2)

#*STEP 2. Solve the transformed equation (2) by the Diagonal Sum Method, It proceeds by composing factor pairs of a*c following these 3 Tips:

*TIP 1. When roots have opposite signs, compose factor pairs of c with all first numbers being negative.

*TIP 2. When both roots are negative, compose factor pairs of a*c with all negative numbers.

*TIP 3. When both roots are positive, compose factor pairs of a*c with all positive numbers.

#*STEP 3. Suppose the 2 obtained real roots of the transformed equation (2) are y1, and y2. Divide both y1 and y2 by the constant (a) to get the 2 real roots x1 and x2 of the original equation.

#*Example 1. Solve: 6x^2 - 19x - 11 = 0. Transformed equation: x^2 - 19x - 66 = 0 (2). Solve equation (2) by the Diagonal Sum Method. Roots have different signs. Compose factor pairs of a*c = -66. Proceeding: (-1, 66) (-2, 33)(-3, 22). This sum is -3 + 22 = 19 = -b. The 2 real roots are y1 = -3 and y2 = 22. Next, find the 2 real roots of the original equation (1): x1 = y1/6 = -3/6 = -1/2, and x2 = y2/2 = 22/6 = 11/3.

#*Example 2. Solve: 16x^2 - 62x + 21 = 0 (1). Transformed equation: x^2 - 62x + 336 = 0, (2). Both roots are positive. Compose factor pairs of a*c = 336. Proceeding: (1, 336)(2, 168)(4, 82)(6, 56). This last sum is 62 = -b. The 2 real roots are: y1 = 6 and y2 = 56. Next, find: x1 = y1 = 16 = 6/16 = 3/8, and x2 = y2/16 = 56/16 = 7/2.

Please read articles titled: "Solving quadratic equations by the new Transforming Method" on Google or Yahoo Search.

The first choice is the quadratic formula since it can solve any quad. equation in standard form ax^2 +bx + c = o, when you can use calculators. However, there are some inconveniences and exceptions:

- You must remember by heart the quadratic formula.

- You must know how to extract the square root of a number when calculators are not allowed, in some test/exams for example.

- In some cases, other solving methods, such as the factoring AC Method and the new Diagonal Sum Method, are simpler and faster.

- Teaching instructions want you to solve quad. equations by other methods in order to improve your math skills.

When a quad. equation can be factored, there are 2 best methods to solve it.

1. The factoring AC Method that proceeds to factor the equations into 2 binomials in x. Then, it solves the 2 binomials for x.

Note. There is a "new and improved factoring AC Method" that was recently introduced by Nghi H Nguyen, the author (Yahoo Search or Google Search). This improved methods helps to solve quad. equations simpler and faster.

2. The new Diagonal Sum Method that directly finds the 2 real roots in the form of 2 fractions, knowing their sum (-b/a) and their product (c/a). You can read this new method in math articles titled:"Solving quadratic equations by the Diagonal Sum Method" (Yahoo Search or Google Search)

0=xa

^{2+xb+c}I may turn this into an instructable at some point...If you are not completely comfortable with solving linear equations, please stop and review that first. It's much better to realize that you need practice with steering when you're in a car than when you're in a fighter jet.

My personal recommendation for solving quadratic equations is to start with the standard form and turn it into the factored form using three memorized shortcuts. I also suggest keeping your algebra readable by working on graph paper and neatly writing one symbol or number per box. Write a sign on every number.

Start by writing the equation in the standard form, y = ax

^{2}+bx +c.Write a=, b=, and c= on separate lines and fill in those numbers.

Write the formula h = -b over 2a and calculate h.

Write the formula k = c -ah

^{2}and calculate k.Write the formula z = sqrt( -k over a ) and calculate z.

Once you have c, h, k, and z, you can easily find anything. You can write the equation in factored form y = a(x -h)

^{2}+k or you can graph and solve it:Solutions: x=h+z or x=h-z

X-intercepts, if z is real: (h+z,0) and (h-z,0)

Y-intercept: (0,c)

Line of symmetry: (y=h)

Vertex: (h,k)

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Here are five illustrations of the steps for an example problem:

Set up problem

Find horizontal part (h)

Find vertical part (k)

Find root/vertex distance (z)

Find solutions (x)

^{2}+bx +c or in the factored form y = a(x -h)^{2}+k . When you are asked to solve them, this usually means setting Y=0 and solving for X = the roots (also known as X-intercepts or zeroes). There are many methods of solving. Some of them are the quadratic formula, completing the square, graphing, and factoring.The best methods to use on y = ax

^{2}+bx +c types:*The quadratic formula is hard to memorize, but when you have it, solving is easy.

*Completing the square is terribly tedious, but its step-by-step nature helps you understand what is going on.

The best methods to use on y = a(x -h)

^{2}+k types:*Graphing just involves plotting points. It's tedious and imprecise, but not too hard.

*Factoring can be used to split your equation into two easier-to-solve linear equations. It doesn't always work.

Unfortunately, understanding why the methods work is considerably more trouble than actually using them. This may be why your teacher said, "here, memorize this". For example, the quadratic formula was built by completing the square.

the best answer, but that option is not available to areplyI may turn this into an instructable at some point...That would be GREAT (I am sorry I didn't get this asked in time to get it included in the MATH Burning Questions contest).

Had you posted this as a separate thread here, I could have marked it as the

Best AnswerThank you for taking the time to explain this: I will run through it more carefully later, and ask any questions I may have.