Find the points of intersection of parabola Y=x^2-7x+23 and the straight line Y=7x-1.
using substitution,
x^2 - 7x + 23 = 7x - 1
x^2 - 14x + 24 = 0
(x - 12)(x - 2) = 0
x = 12 or x = 2
if x=12, y = 7(12)-1 = 83
if x=2 , y = 7(2)-1 = 13
they intersect at (12,83) and (2,13)
confirmation here:
look at the first graph
http://www.wolframalpha.com/input/?i=plot+Y%3Dx%5E2-7x%2B23+%2C+Y%3D7x-1+from+0+to+15
Yes?
To find the points of intersection between the parabola \(y = x^2 - 7x + 23\) and the line \(y = 7x - 1\), we can set the two equations equal to each other:
\(x^2 - 7x + 23 = 7x - 1\)
Let's solve this quadratic equation step by step.
Step 1: Rearrange the equation:
\(x^2 - 7x - 7x + 23 + 1 = 0\)
Simplifying, we have:
\(x^2 - 14x + 24 = 0\)
Step 2: Factor the quadratic equation:
\((x - 2)(x - 12) = 0\)
Using the zero product property, we set each factor equal to zero and solve for \(x\):
\(x - 2 = 0\) or \(x - 12 = 0\)
Step 3: Solve for \(x\):
\(x = 2\) or \(x = 12\)
So, the points of intersection between the parabola and the line are when \(x = 2\) and \(x = 12\).
Step 4: Substitute the values of \(x\) back into either equation to find the corresponding \(y\) values:
When \(x = 2\):
\(y = 7(2) - 1\)
\(y = 14 - 1\)
\(y = 13\)
When \(x = 12\):
\(y = 7(12) - 1\)
\(y = 84 - 1\)
\(y = 83\)
So, the two points of intersection are \((2, 13)\) and \((12, 83)\).
To find the points of intersection between a parabola and a straight line, we need to set the equation of the parabola equal to the equation of the straight line and solve for the x-coordinate(s) of the intersection point(s).
Given:
Parabola: y = x^2 - 7x + 23
Straight line: y = 7x - 1
Step 1: Set the equations equal to each other:
x^2 - 7x + 23 = 7x - 1
Step 2: Rearrange to a quadratic equation:
x^2 - 7x - 7x + 23 + 1 = 0
x^2 - 14x + 24 = 0
Step 3: Solve the quadratic equation using factoring, completing the square, or the quadratic formula.
Since the quadratic equation cannot be easily factored, we'll use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a), where a = 1, b = -14, and c = 24.
x = (-(-14) ± √((-14)^2 - 4(1)(24))) / (2(1))
x = (14 ± √(196 - 96)) / 2
x = (14 ± √100) / 2
x = (14 ± 10) / 2
Now, we have two possible x-coordinates of the intersection points:
Case 1: x = (14 + 10) / 2 = 24 / 2 = 12
Case 2: x = (14 - 10) / 2 = 4 / 2 = 2
Step 4: Substitute the x-values back into either of the original equations to find the corresponding y-values.
In this case, we'll use the straight line equation y = 7x - 1.
For case 1 (x = 12):
y = 7(12) - 1
y = 84 - 1
y = 83
So, the first point of intersection is (12, 83).
For case 2 (x = 2):
y = 7(2) - 1
y = 14 - 1
y = 13
So, the second point of intersection is (2, 13).
Therefore, the points of intersection of the parabola y = x^2 - 7x + 23 and the straight line y = 7x - 1 are (12, 83) and (2, 13).