Find all solutions to the following system of equations x+2y=0 ; x²-2x+y²-2y-3=0. illustrate with a graph.

The solutions to the system of equations are the points (2, -1) and (-1.2, 0.6)

Step-by-step explanation:

Hi there!

We have the following system of equations:

x + 2y = 0

x² - 2x + y²- 2y - 3 = 0

The solutions of the system are the pairs (x, y) that satisfy both equations.

Let´s take the first equation and solve it for x:

x + 2y = 0

x = -2y

And now let´s replace the x in the second equation:

x² - 2x + y²- 2y - 3 = 0

(-2y)² -2(-2y) + y² - 2y - 3 = 0

4y² +4y + y² - 2y - 3 = 0

5y² + 2y - 3 = 0

Let´s use the quadratic formula to solve this quadratic equation:

[-b ± √(b² - 4ac)]/2a

a = 5

b = 2

c = -3

The solutions to this equation are:

y = -1 and y = 0.6

The values of x will be:

x = -2y

x = -2(-1) = 2

and

x =-2(0.6) = -1.2

The solutions to the system of equations are the points (2, -1) and (-1.2, 0.6)

Please, see the attached figure. The points where the curves intersect are the solutions of the system.

Have a nice day!

answer is 8x+24

area of vanessa's patio = length × width

given, length = (3x² + 5x + 10)

width = (x² - 3x - 1)

area of vanessa's patio = (3x² + 5x + 10)(x² - 3x - 1)

= 3x²(x² - 3x - 1) + 5x(x² - 3x - 1) + 10(x² - 3x - 1)

= 3x⁴ - 9x³ - 3x² + 5x³ - 15x² - 5x + 10x² - 30x - 10

= 3x⁴ - 4x³ - 8x² - 35x - 10

hence, expression of vanessa's patio = (3x⁴ - 4x³ - 8x² - 35x - 10)