Task 1
Nonlinear Systems of Equations

Create a system of equations that includes one linear equation and one quadratic equation.

Part 1. Show all work in solving your system of equations algebraically.

Part 2. Graph your system of equations and show the solution graphically to verify your solution.

Task 2
Polynomial Identities

Part 1. Pick a two-digit number greater than 25. Rewrite your two-digit number as a difference of two numbers. Show how to use the identity (x − y)2 = x2 − 2xy + y2 to square your number without using a calculator.

Part 2. Choose two values, a and b, each between 8 and 15. Show how to use the identity a3 + b3 = (a + b)(a2 − ab + b2) to calculate the sum of the cubes of your numbers without using a calculator.

See procedure below;

Step-by-step explanation:

Part 1. Take a look at the example of the system below;

y = 2x^2 - x + 1,

y = 4x + 4

Now let us solve through making the two equations equivalent;

2x^2 - x + 1 = 4x + 4,

2x^2 - x - 4x + 1 - 4 = 0,

2x^2 - 5x - 3 = 0

Let us factor and solve for the point of intersection;

2x^2 - 5x - 3 = 0,

( 2x + 1 )( x - 3 ) = 0,

x - 3 = 0, and 2x + 1 = 0;

x = 3, and x = - 1/2

To substitute each value of x into each equation;

y = 2 * ( 3 )^2 - 3 + 1, ⇒ y = 18 - 3 + 1, y = 16,

y = 4 * ( - 1/2 ) + 4, y = - 2 + 4, y = 2

Solution: ( - 1/2, 2 ), and ( 3, 16 )

Part 2. I can't express this graph here, so please do a google search of the two equations y = 2x^2 - x + 1 and y = 4x + 4, you will get the graphical representation you inquire!

Task 2; Part 1. Let us say this two digit number is 30. Now this can be represented as ( 70 - 40 ) say. Let us square this value and represent it in the form provided;

( 70 - 40 )^2,

70^2 - 2 * 70 * 40 + 40^2,

4900 - 5600 + 1600,

- 700 + 1600,

900

Task 2; Part 2. Let us take a to be ⇒ 10, and b ⇒ 12. Let us say this is represented by 10^3 + 12^3, and now compute through the form provided;

10^3 + 12^3,

( 10 + 12)( 10^2 - 10 * 12 + 12^2),

( 22 )( 100 - 120 + 144 ),

( 22 )( 124 ),

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