Pythagorean Quadratic

Pythagorean Quadratic

 

A diagram showing the location of Castle Rock

2x + 4

CASTLE ROCK

x

2x + 6

Using Pythagorean Theorem we know that:

a² + b² = c²

Where; a = x, b = 2x+4 and c = 2x+6

 

Therefore, our Pythagorean Theorem equation will be as shown below:

x² + (2x+4)² = (2x+6)²

The above will be solved as follows;

x² + (2x+4) (2x+4) = (2x+6) (2x+6)

x² + 4x² + 8x+ 8x+ 16 = 4x² +12x+ 12x+ 36

5x² + 16x + 16 = 4x² +24 + 36

 

Simplifying the above compound equation gives;

x² – 8x -20

 

Using zero factor the above polynomial equation can be expressed as a quadratic equation as show below;

x² – 8x -20 = 0

However, using AC method of factoring, the above quadratic equation can be factored following the steps below:

Step 1: Determining AC by multiplying A and C terms

In x² – 8x – 20 A = 1, B = -8, C = -20

AC = 1* -20 = -20

Step 2: Finding factors of AC whose sum equals to B (-8)

The factors of AC (-20) are as follows:

-1 and 20

-20 and 1

-10 and 2

-2 and 10

-5 and 4

-4 and 5

Therefore, factors of AC (-20) whose sum is B (-8) are 2 and -10

Step 3: The original quadratic equation is rewritten as follows:

x² +2x -10x – 20 = 0

Step 4: Factoring the new equation by grouping using the GCF

Where; the perfect square of x and x is x² and the prime factor is x+2

x(x+2) – (-) 10 (x+2)

Step 5: Final factoring by the distributive property

By the distributive property we know that x(x+2) – (-) 10 (x+2) is equivalent to (x-10) (x+2)

The factoring answer is (x-10) (x+2) = 0

 

In order to get the value of x we can solve for x using The Zero Factor Principle

Thus, since (x-10) (x+2) = 0

Then, either (x-10) or (x+2) should be zero as shown in the equations below:

(x – 10) = 0      or      (x+2) = 0

Hence the value of x will either be:

x= 10     or         x= -2

 

However, for this problem the solution where x = 10 is the best solution since when applied to the quadratic equation it is the one which gives more appropriate and realistic values.

 

Finally, translation of the polynomial measurements of the triangle to paces according to the obtained solution is as follows;

Since x = 10 (taken because it is the best solution) the paces are as shown below;

x paces = 10 paces

2x+4 paces = 2×10 + 4 = 24 paces

2x+6 paces = 2×10 + 6 = 26 paces

 

Reference

Dugopolski, M. (2012). Elementary and intermediate algebra, (4^th ed.). New York, NY: McGraw-Hill Publishing.

 

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