Find the points where the parabola y=x+4x+3 intersects with the straight line y=-x+9
To find the points of intersection between the parabola y=x^2+4x+3 and the line y=-x+9, we need to set them equal to each other and solve for x.
Setting the equations equal, we have:
x^2 + 4x + 3 = -x + 9
Combining like terms:
x^2 + 5x + 6 = 0
Now, we can factor the quadratic equation:
(x + 2)(x + 3) = 0
Setting each factor equal to zero:
x + 2 = 0 or x + 3 = 0
Solving for x in each equation:
x = -2 or x = -3
Therefore, the parabola y=x^2+4x+3 intersects the line y=-x+9 at x=-2 and x=-3.
To find the corresponding y-values, substitute these x-values back into either equation:
For x = -2:
y = -(-2) + 9 = 2 + 9 = 11
For x = -3:
y = -(-3) + 9 = 3 + 9 = 12
Therefore, the points of intersection are (-2, 11) and (-3, 12).
x^2 + 5x - 6 = 0 because 3 - 9 = -6
(x-1)(x+6) = 0
x = 1 or x = -6
etc
Apologies for the mistake in the previous response. You are correct.
Let's solve the equation correctly this time.
Setting the equation x^2 + 5x - 6 = 0.
We can factor the quadratic equation as:
(x - 1)(x + 6) = 0.
Setting each factor equal to zero:
x - 1 = 0 or x + 6 = 0.
Solving for x in each equation:
x = 1 or x = -6.
Therefore, the parabola y = x^2 + 4x + 3 intersects the line y = -x + 9 at x = 1 and x = -6.
To find the corresponding y-values, substitute these x-values back into either equation:
For x = 1:
y = -1 + 9 = 8.
For x = -6:
y = 6 + 9 = 15.
Therefore, the points of intersection are (1, 8) and (-6, 15).
To find the points where the parabola and the straight line intersect, we need to solve the system of equations formed by equating the two equations.
Let's set the equations equal to each other:
x^2 + 4x + 3 = -x + 9
Rearranging the equation, we get:
x^2 + 5x - 6 = 0
Now we can solve this quadratic equation by factoring or using the quadratic formula. I'll use the quadratic formula to find the values of x:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
In this equation, a = 1, b = 5, and c = -6. Substituting these values, we get:
x = (-5 ± sqrt(5^2 - 4(1)(-6))) / (2*1)
x = (-5 ± sqrt(25 + 24)) / 2
x = (-5 ± sqrt(49)) / 2
x = (-5 ± 7) / 2
Now, we can find the corresponding y-values by substituting these x-values into either of the given equations. Let's use the equation y = -x + 9:
For x = (-5 + 7) / 2 = 1:
y = -(1) + 9 = 8
For x = (-5 - 7) / 2 = -6:
y = -(-6) + 9 = 15
So, the points of intersection are (1, 8) and (-6, 15).