Question 177920
this equation doesn't have any roots.
when you solve for the roots you get a complex number.
complex number means there are no roots.
here's how i did it.
your equation is:
{{{(x/-7) = (x+1)/(x-11)}}}
muiltiply both sides by -7 to get:
{{{x = -7 * (x+1)/(x-11)}}}
multiply both sides by (x-11) to get:
{{{x * (x-11) = -7 * (x+1)}}}
simplify to get:
{{{x^2 - 11x = -7x - 7}}}
add 7x to both sides to get:
{{{x^2 - 11x + 7x = -7}}}
add 7 to both sides to get":
{{{x^2 - 11x + 7x + 7 = 0}}}
simplify to get:
{{{x^2 - 4x + 7 = 0}}}
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you can solve this using the quadratic formula or completing the squares.
either way will get you the same answer.
i'll use the quadratic formula first:
a = 1 = coefficient of the x^2 term.
b = -4 = coefficient of the x term.
c = 7 = constant term.
quadratic formula is:
{{{x = (-b +- sqrt(b^2-4ac)/(2a))}}}
substituting values for a,b,c gets:
{{{x = (-(-4) +- sqrt((-4)^2 - 4*7*1)/(2*1))}}}
this becomes:
{{{x = 4 +- sqrt(16 - 28)/(2)}}}
which becomes:
{{{x = 4 +- sqrt(-12)/(2)}}}
which becomes:
{{{x = 4 +- sqrt(-3*2*2) / (2) }}}
which becomes:
{{{ x = 4 +- 2*sqrt(-3)/(2)}}}
which becomes:
{{{ x = 2 +- sqrt(-3)}}}
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you can also solve this by completing the squares as follows:
your equation after manipulating it is:
{{{x^2 - 4x + 7 = 0}}}
this is the same place where we solved it using the quadratic formula.
this time, however, we will complete the squares.
subtract 7 from both sides to get:
{{{x^2 - 4x = -7}}}
{{{x^2 - 4x = (x-2)^2 - 4}}}
your equation becomes:
{{{(x-2)^2 - 4 = -7}}}
you add 4 to both sides to get:
{{{(x-2)^2 = -3}}}
you take the square root of both sides to get:
{{{x-2}}} = +/- {{{sqrt(-3)}}}
you add 2 to both sides to get:
{{{x = 2 +- sqrt(-3)}}}
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since the value of x is a complex number, this means that there are no roots to the equation.
since the roots are when the equation crosses the x-axis, this means that this equation does not cross the x-axis.
a graph of {{{x^2-4x+7}}} is shown below:
{{{graph(800,800,-20,20,-20,20,x^2-4x+7)}}}