1. Solve the system of equations by elimination.y=2x^2-2x-3 and y=-x^2-2x-3
2.The system of equations y=-5(x+4)^2-4 and y=8x^2+64x+124 has _____ solution(s).
3. Determine the value of k in y=kx^2-5x+2 that will result in the intersection of the line y=-3x+4 with the quadratic at
a) two points (1 mark)
b) one points (1 mark)
c) no point (1 mark)
#1 eliminate y by equating the two:
2x^2-2x-3 = -x^2-2x-3
Now solve for x, and then you can get y
#2 same thing
#3 again, use the discriminant. We want
kx^2-5x+2 = -3x+4
kx^2-2x-6 = 0
The discriminant is 4+24k
1. To solve the system of equations by elimination, we need to eliminate one of the variables by adding or subtracting the equations. In this case, we can eliminate the variable "y" by subtracting one equation from the other.
Given equations:
y = 2x^2 - 2x - 3 (Equation 1)
y = -x^2 - 2x - 3 (Equation 2)
To eliminate "y", we subtract Equation 2 from Equation 1:
(2x^2 - 2x - 3) - (-x^2 - 2x - 3) = 0
Simplify the equation:
2x^2 - 2x - 3 + x^2 + 2x + 3 = 0
Combine like terms:
3x^2 = 0
Divide both sides by 3:
x^2 = 0
Take the square root of both sides:
x = 0
Now substitute the value of "x" back into either Equation 1 or 2 to find the corresponding value of "y". Let's use Equation 1:
y = 2(0)^2 - 2(0) - 3
y = -3
Therefore, the solution to the system of equations is x = 0 and y = -3.
2. To determine the number of solutions for the given system of equations, we need to compare the two equations and analyze their behavior.
Given equations:
y = -5(x + 4)^2 - 4 (Equation 1)
y = 8x^2 + 64x + 124 (Equation 2)
Looking at Equation 1, we can see that it represents a parabola with its vertex at the point (-4, -4) and it opens downward. This means that it is a maximum point.
Looking at Equation 2, we can see that it represents another parabola. Since the coefficient of x^2 is positive, it opens upward. This means that it is a minimum point.
Since the maximum point of Equation 1 is below the minimum point of Equation 2, the two parabolas do not intersect. Therefore, there are no solutions to the system of equations.
3. To determine the value of "k" in the quadratic equation y = kx^2 - 5x + 2 that will result in the intersection of the line y = -3x + 4 with the quadratic, we need to set the two equations equal to each other and solve for "x".
Given equations:
y = kx^2 - 5x + 2 (Equation 1)
y = -3x + 4 (Equation 2)
Setting the equations equal to each other:
kx^2 - 5x + 2 = -3x + 4
Rearranging the equation:
kx^2 - 5x + 2 + 3x - 4 = 0
Combine like terms:
kx^2 - 2x - 2 = 0
Now, depending on the discriminant (b^2 - 4ac), we can determine the number of intersection points between the two equations.
a) For two intersection points, the discriminant should be greater than 0:
(-2)^2 - 4(k)(-2) > 0
4 + 8k > 0
8k > -4
k > -0.5
Therefore, for two intersection points, the value of k should be greater than -0.5.
b) For one intersection point, the discriminant should be equal to 0:
(-2)^2 - 4(k)(-2) = 0
4 + 8k = 0
8k = -4
k = -0.5
Therefore, for one intersection point, the value of k should be -0.5.
c) For no intersection points, the discriminant should be less than 0:
(-2)^2 - 4(k)(-2) < 0
4 + 8k < 0
8k < -4
k < -0.5
Therefore, for no intersection point, the value of k should be less than -0.5.
In summary:
a) k > -0.5 for two intersection points.
b) k = -0.5 for one intersection point.
c) k < -0.5 for no intersection point.