(QUADRATIC FUNCTIONS)

Question

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.

Answer 1

y=0

x^2 + y^2 = 4

x^2 + 0 = 4

x =+/-sqr4 = + or -2

4 + y^2 = 4

y^2 = 0

y = 0

solutions are (-2,0) and (2,0)

graphically the quadratic is a circle with center at the origin and radius = 2

the linear equation is a flat horizontal line = the x axis, which cuts the circle in half

intersecting the circle at its right and left sides

30^2 = 900

(x-y)^2 = x^2 -2xy +y^2 = (35-5)^2 = 35^2 -2(35)(5) + 5^2

= 1225 -350 +25 = 1250-700 = 900

9^2 +10^2 = 729+1000 = 1729

=(a+b)(a^2 -ab + b^2)

= (9+10)(9^2 -9(10) + 10^2)

= 19(81 -90 +100)

= 19(91)= 1729