SOLUTION: Find the x-intercept(s), if any, of the parabola given by the function f(x) = -x^2 + x - 1.

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equation Customizable Word Problems -> SOLUTION: Find the x-intercept(s), if any, of the parabola given by the function f(x) = -x^2 + x - 1.      Log On


   

Question 356311: Find the x-intercept(s), if any, of the parabola given by the function f(x) = -x^2 + x - 1.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29+=+-x%5E2+%2B+x+-1
X-intercepts are points where a graph intercepts/interesects the x axis. All points on the x axis have a y coordinate that is zero. (Think about it.) This tells us that X-intercepts have y coordinates that are zero. And since f(x) represents the y value, we want f(x) to be zero:
0+=+-x%5E2+%2B+x+-1

This is a quadratic equation. To solve a quadratic equation we first get one side of the equation to be zero. (We already have one side zero!). Then you either factor or use the Quadratic Formula. -x%5E2+%2B+x+-1 does not factor easily we will use the Quadratic Formula:
x+=+%28-b+%2B-+sqrt%28b%5E2+-4ac%29%29%2F%282a%29
where the "a", "b" and "c" come from the general quadratic equation:
ax%5E2+%2B+bx+%2B+c+=+0
Looking at your equation we can see that
a = -1
b = 1
c = -1 (Be careful here. c = -1 because the general form has "+ c" at the end so we have to look at your equation as if there were all additions. So we have to look at " - 1" as if it were "+ (-1)". This is why c = -1.)
Subsituting these values for a, b and c into the formula we get:
x+=+%28-%281%29+%2B-+sqrt%28%281%29%5E2+-+4%28-1%29%28-1%29%29%29%2F%282%28-1%29%29
(Note how I used parentheses when I substituted. This is a very good habit to develop. It helps you avoid many kinds of errors that occur when you neglect to use parentheses.)
Simplifying we get:
x+=+%28-1+%2B-+sqrt%281+-+4%28-1%29%28-1%29%29%29%2F%28-2%29
x+=+%28-1+%2B-+sqrt%281+-+4%29%29%2F%28-2%29
x+=+%28-1+%2B-+sqrt%28-3%29%29%2F%28-2%29
At this point we should notice a problem. We have sqrt%28-3%29 in our solution. But there are no Real numbers which can be squared to get -3. This means that there are no solutions to 0+=+-x%5E2+%2B+x+-1. And since there are no x values that make the y value 0, there are no points on the x axis (i.e. no x-intercepts). IOW, this parabola does not cross the x axis.

Here's a graph so you can see this:
graph%28400%2C+400%2C+-4%2C+4%2C+-4%2C+4%2C+-x%5E2%2Bx-1%29