SOLUTION: Use substitution with the two equations and determine the points of intersection. Show all work. y=x^2-1 y=x

Algebra ->  Graphs -> SOLUTION: Use substitution with the two equations and determine the points of intersection. Show all work. y=x^2-1 y=x      Log On


   

Question 228283: Use substitution with the two equations and determine the points of intersection. Show all work.
y=x^2-1
y=x
Answer by drj(1380) About Me  (Show Source):
You can put this solution on YOUR website!
Use substitution with the two equations and determine the points of intersection. Show all work.

y=x%5E2-1 Equation A
y=x Equation B

Step 1. Substitute Equation B into Equation A

x=x%5E2-1

Subtract x from both sides of the equation

x-x=x%5E2-1-x

0=x%5E2-x-1

Step 2. To solve, use quadratic formula given as

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+

where a=1, b=-1 and c=-1.

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-1x%2B-1+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-1%29%5E2-4%2A1%2A-1=5.

Discriminant d=5 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--1%2B-sqrt%28+5+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-1%29%2Bsqrt%28+5+%29%29%2F2%5C1+=+1.61803398874989
x%5B2%5D+=+%28-%28-1%29-sqrt%28+5+%29%29%2F2%5C1+=+-0.618033988749895

Quadratic expression 1x%5E2%2B-1x%2B-1 can be factored:
1x%5E2%2B-1x%2B-1+=+1%28x-1.61803398874989%29%2A%28x--0.618033988749895%29
Again, the answer is: 1.61803398874989, -0.618033988749895. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-1%2Ax%2B-1+%29



Step 3. ANSWER: One solution is x=1.618 and y=1.618 or (1.618, 1.618) and the other solution is x=-0.618 and y=-0.618 or (-0.618,-0.618)

Here's a graph between the the line y=x and parabola y=x%5E2-1:

graph%28400%2C400%2C-5%2C5%2C-5%2C+5%2C+x%2C+x%5E2-1%29

I hope the above steps were helpful.

For FREE Step-By-Step videos in Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra and for Trigonometry visit http://www.FreedomUniversity.TV/courses/Trigonometry.

And good luck in your studies!

Respectfully,
Dr J
http://www.FreedomUniversity.TV