document.write( "Question 356311: Find the x-intercept(s), if any, of the parabola given by the function f(x) = -x^2 + x - 1. \n" ); document.write( "

Algebra.Com's Answer #254427 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"f%28x%29+=+-x%5E2+%2B+x+-1\"$
\n" ); document.write( "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:
\n" ); document.write( " $\"0+=+-x%5E2+%2B+x+-1\"$

\n" ); document.write( "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:
\n" ); document.write( " $\"x+=+%28-b+%2B-+sqrt%28b%5E2+-4ac%29%29%2F%282a%29\"$
\n" ); document.write( "where the \"a\", \"b\" and \"c\" come from the general quadratic equation:
\n" ); document.write( " $\"ax%5E2+%2B+bx+%2B+c+=+0\"$
\n" ); document.write( "Looking at your equation we can see that
\n" ); document.write( "a = -1
\n" ); document.write( "b = 1
\n" ); document.write( "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.)
\n" ); document.write( "Subsituting these values for a, b and c into the formula we get:
\n" ); document.write( " $\"x+=+%28-%281%29+%2B-+sqrt%28%281%29%5E2+-+4%28-1%29%28-1%29%29%29%2F%282%28-1%29%29\"$
\n" ); document.write( "(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.)
\n" ); document.write( "Simplifying we get:
\n" ); document.write( " $\"x+=+%28-1+%2B-+sqrt%281+-+4%28-1%29%28-1%29%29%29%2F%28-2%29\"$
\n" ); document.write( " $\"x+=+%28-1+%2B-+sqrt%281+-+4%29%29%2F%28-2%29\"$
\n" ); document.write( " $\"x+=+%28-1+%2B-+sqrt%28-3%29%29%2F%28-2%29\"$
\n" ); document.write( "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.

\n" ); document.write( "Here's a graph so you can see this:
\n" ); document.write( " $\"graph%28400%2C+400%2C+-4%2C+4%2C+-4%2C+4%2C+-x%5E2%2Bx-1%29\"$
\n" ); document.write( " \n" ); document.write( "

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