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Factoring a Quadratic Expression











































Factoring a quadratic

v² - 43v - 728 = 0

The expression on the left side of the equal sign is called a quadratic polynomial.

The entire equation (including the equal sign and the 0) is called a quadratic equation.

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May 03, 2013
Factoring a quadratic
by: Staff


Answer

Part I


There are many ways to factor the expression.

I'm going to demonstrate three ways.


Completing the square
Quadratic formula


Completing the square and using the quadratic formula to obtain the factors will be a direct, forthright calculation.


Trial and Error (there are many different approaches)


The standard approach to factoring simple quadratic equations is the trial and error method. However, because your equation contains large numbers (-43 and -728), using the trial and error method will be more involved. For this reason I am going to cover this approach last.


Completing the Square to obtain factors

Completing the square will allow you to compute the factors directly. No trial and error is involved.



To complete the square, begin with the quadratic equation in this format:


Quadratic Equation - variable v




For your equation:

Quadratic Equation - coefficient values





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May 03, 2013
Factoring a quadratic
by: Staff


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Part II


add the constant (the number 728) to each side of the equation to move the coefficient c to the right side of the equation


Complete the Square - add the constant (the number 728) to each side of the equation





Divide the coefficient b (which is -43) by 2

Divide the Coefficient b by 2





Square the result

Divide the Coefficient b by 2, and then square the result





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May 03, 2013
Factoring a quadratic
by: Staff


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Part III


Add this value to each side of the equation. This means that you will add 1849/4 (or it's equivalent 43²/2²) to each side of the equation.

Add the result to each side of the equation - this will CHANGE THE LEFT SIDE OF THE EQUATION SO IT CAN BE FACTORED AS A PERFECT SQUARE






THE REASON for adding 1849/4 (or it's equivalent 43²/2²) to each side of the equation is: CHANGE THE LEFT SIDE OF THE EQUATION SO IT CAN BE FACTORED AS A PERFECT SQUARE:

Factor the left side of the equation - it is a perfect square






Take the square root of each side of the equation

Take the square root of each side of the equation






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May 03, 2013
Factoring a quadratic
by: Staff


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Part IV


Add 43/2 from each side of the equation to remove the 43/2 from the left side of the equation. This leaves the variable v as the only term on the left side of the equation.

Add 43/2  from each side of the equation to remove the 43/2  from the left side of the equation.  This leaves the variable v as the only term on the left side of the equation.




1st value, v₁

1st value,  v₁




2nd value, v₂

2nd value, v₂






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May 03, 2013
Factoring a quadratic
by: Staff


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Part V

The final solution to the quadratic equation is:

Final solution set for quadratic equation





To find the factors for the original equation, reverse the signs:

Reverse the signs to determine the quadratic factors





the final answer is:

Factoring the quadratic:  final factors





Using the Quadratic formula to obtain factors

Using the Quadratic Formula is a short cut to Completing the Square. No trial and error is involved.

Using the Quadratic Formula is a short cut to Completing the Square.  No trial and error is involved.







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May 03, 2013
Factoring a quadratic
by: Staff

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Part VI

Solving for the unknown variable using the quadratic formula.





Solution for v using the quadratic formula.




From this point on [v = (43 ± 69)/2], the steps needed to determine the factors are exactly what they were when completing the square. Refer to the final steps shown under "Completing the Square", above.



Using Trial and Error to obtain factors


This is the standard approach to factoring simple quadratic equations. For equations with small coefficients, trial and error is fast and efficient.

In this case it is not the ideal way to factor the equation. This equation contains a large coefficient (-43) and a large constant (-728).


1. List all the factors for the constant "c" (the number -728) and then list all the factor combinations which can be multiplied to = c

Factors of the number -728, the constant C in the quadratic equation





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May 03, 2013
Factoring a quadratic
by: Staff


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Part VII

Factors of the number -728, the constant C in the quadratic equation, part 1




Factors of the number -728, the constant C in the quadratic equation, part 2





3. Write down two sets of empty parentheses

Factoring by Trial and Error - begin by writing down two sets of empty parentheses





4. On left side of each set of parentheses, write the variable v.

Factoring - On left side of each set of parentheses, write the variable v.







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May 03, 2013
Factoring a quadratic
by: Staff


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Part VIII


5. Add each of the two factors in step 1. Find which two factors can be added to produce a total equal to the coefficient b (-43).

 Find two factors which can be added to produce a total equal to the coefficient b, -43, part 1




Find two factors which can be added to produce a total equal to the coefficient b, -43, part 2





As you can see, only two of the possible factors can be added to produce the value of -43 (the coefficient "b"). They are +13 and -56.

These are the missing values which must put in the parentheses of step 4.

The trial and error technique has determined the following factors to the quadratic equation:

The trial and error technique has determined the following factors to the quadratic equation








Thanks for writing.

Staff
www.solving-math-problems.com



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