Question

100 POINTS !!!PLEASE HELP I NEED TO PASS THIS !!! WILL MARK BRANLIEST FOR THE BEST ANSWER. please answer to the fullest and elaborate thoroughly 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. thank you so much of you helped thos is really hard for me and i need to pass.n

Answer 1

Hello, please consider the following.

Task 1 - solve the following system

$\begin{cases} y = 2x+4 &\text{** linear equation **} \\ y = x^2+6x+8 &\text{** quadratic equation **} \end{cases}$

Part 1 - algebraically

y=y, right? So, we can write.

We will come up with an equation to solve, that we can solve by completing the square, or using the discriminant, or factorising knowing the sum and the product of the zeroes.

$x^2+6x+8=2x+4\\\\x^2+6x+8-2x-4=0\\\\x^2+4x+4=x^2+2*2*x+2^2=(x+2)^2=0\\\\\text{** we take the root **}\\\\x+2=0 \ \ so\ \ \boxed{x = -2}$

Part 2 - graphically

I attached the graph of the line and the parabola.

And then we check for the intersection points.

Here, we can see that there is one point of intersection for x = -2 and then y = 0.

Task 2 - Polynomial Identities.

Part 1 - Square

Let's take 30=50-20=10(5-2)

$30^2=(50-20)^2\\\\=10^2(5-2)^2\\\\=100\left( 5^2-2*2*5+2^2\right)\\\\=100*(25-20+4)\\\\=100*(9)\\\\=900$

Using this formula, we just need to compute smaller squares, so it helps for the computation.

Part 2 - Cubic

Let's take 9 and 14.

$9^3+14^3=(9+14)(9^2-9*14+14^2)\\\\=23*(81-126+196)\\\\=23*(277-131)\\\\=23*151\\\\=3473$

This is the same idea here where we have to compute smaller numbers.

Thank you