Determine the coordinates of the point (xy) where the curves y=(12)x−5 and y=x2+2x−15 intersect in the third quadrant
y = 12x - 5. y = x^2 + 2x - 15.
Substitute 12x - 5 for y in the 2nd Eq:
12x -5 = x^2 + 2x -15,
Solve for x,
12x - x^2 - 2x = -15 + 5,
Combine like-terms:
-x^2 + 10x + 10 = 0,
Multiply each side by -1:
x^2 - 10x - 10 = 0,
Use quadratic formula to solve for x:
x = (10 +- sqrt(100+40)) / 2,
x = 10.92 , -0.916,
x = -0.916(3rd quadrant).
Substitute -0.916 for x in 1st Eq:
y = 12 * -0.916 - 5 = -16.
P(-0.916 , -16)
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Well, well, well, looks like we've got some curves intersecting in the third quadrant. It's like a clandestine arrangement between numbers. Let's crack this code, shall we?
First, we need to find where these two sneaky curves meet. To do that, we'll set the equations equal to each other:
(12)x - 5 = x^2 + 2x - 15
Now, let's bring everything to one side and attempt to solve this equation:
x^2 + 2x - 12x + 10 = 0
Simplifying, we get:
x^2 - 10x + 10 = 0
Using our handy-dandy quadratic formula (I call it the detective of equations!), we find that:
x = (10 ± √(10^2 - 4 * 1 * 10)) / (2 * 1)
After doing some detective work (also known as math), we find two potential x coordinates: x ≈ 1.44 and x ≈ 8.56 (approximately).
Now, let's find the corresponding y-coordinates. Plug each x value into either original equation, and we'll get our answers:
For x ≈ 1.44:
y ≈ (12 * 1.44) - 5
y ≈ 17.28 - 5
y ≈ 12.28
For x ≈ 8.56:
y ≈ (12 * 8.56) - 5
y ≈ 102.72 - 5
y ≈ 97.72
So, the coordinates where the curves intersect in the third quadrant are approximately (1.44, 12.28) and (8.56, 97.72). Enjoy your clandestine rendezvous with these numbers!
To find the coordinates of the point where two curves intersect, we need to set their equations equal to each other and solve for the values of x and y.
The given curves are:
Curve 1: y = (12)x - 5
Curve 2: y = x^2 + 2x - 15
Setting the equations equal to each other, we have:
(12)x - 5 = x^2 + 2x - 15
Rearranging the equation, we get:
0 = x^2 + 2x - (12)x + 2x - 15 + 5
0 = x^2 - 8x - 10
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
For our equation, a = 1, b = -8, and c = -10. Substituting these values into the quadratic formula, we have:
x = (-(-8) ± sqrt((-8)^2 - 4(1)(-10))) / (2(1))
x = (8 ± sqrt(64 + 40)) / 2
x = (8 ± sqrt(104)) / 2
x = (8 ± √104) / 2
x = 4 ± √26
Since we are looking for the coordinates in the third quadrant, the x-coordinate should be negative. Therefore, x = 4 - √26.
Next, we substitute this value back into one of the original equations to find the y-coordinate. Let's use Curve 1:
y = (12)(4 - √26) - 5
y = 48 - 12√26 - 5
y = 43 - 12√26
Therefore, the coordinates of the point where the two curves intersect in the third quadrant are (x, y) = (4 - √26, 43 - 12√26).
To determine the coordinates of the point of intersection between the two curves, we need to find the values of x and y that satisfy both equations simultaneously. In this case, the equations are y = (12)x - 5 and y = x^2 + 2x - 15.
To find the intersection point, we'll set the two equations equal to each other and solve for x:
(12)x - 5 = x^2 + 2x - 15
Rearranging the equation, we get:
x^2 - 10x + 10 = 0
Now, we can solve this quadratic equation for x using factoring or the quadratic formula. In this case, factoring may be a bit difficult, so we'll use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 1, b = -10, and c = 10. Plugging these values into the formula, we get:
x = (-(-10) ± √((-10)^2 - 4(1)(10))) / (2(1))
= (10 ± √(100 - 40)) / 2
= (10 ± √60) / 2
= (10 ± 2√15) / 2
Simplifying further, we have:
x = 5 ± √15
Since we are looking for a point in the third quadrant, x should be negative. So we take the negative root:
x = 5 - √15
Now, substitute this value of x into either of the original equations to find the corresponding y-coordinate. Let's use the first equation:
y = (12)(5 - √15) - 5
= 60 - 12√15 - 5
= 55 - 12√15
Therefore, the coordinates of the point of intersection in the third quadrant are approximately (x, y) ≈ (5 - √15, 55 - 12√15).