We want to solve the quadratic equation
x squared minus 10x plus 34 equals zero.
Since there are no factors of positive 34
that add to negative 10,
this is not factorable,
and therefore we will use the
quadratic formula to solve.
However, if you look below, I also graphed
the corresponding quadratic function
on the coordinate plane, and notice how
the parabola does not intersect
the horizontal or x-axis, which means
there are no real x values
that make the y value
equal to zero, which means
there are no real solutions
to the given equation, and therefore,
we can expect complex solutions.
Let's begin by listing
the values of a, b, and c,
where a is the coefficient of x squared,
b is the coefficient of x,
and c is the constant term.
Remember, before we determine
the values of a, b, and c,
the equation must be set equal to zero.
So again here, a is equal to one,
b is equal to negative 10,
and c is equal to positive 34.
And now I'll perform substitution
into the quadratic formula.
We have x equals, in the
numerator we have negative b,
which is negative, and then negative 10,
plus or minus the square
root of the quantity
b square, which is the
square of negative 10,
minus four times a times
c, which is minus four
times one times 34.
All this is over two times
a, which is two times one.
Now I begin simplifying.
We have x equals the
opposite of negative 10,
or negative negative 10, is positive 10,
plus or minus the square
root of, the square
of negative 10 is 100, and
then we have minus four
times one times 34, is 136,
and this is all divided
by two times one, which is two.
Simplifying under the
square root, 100 minus 136
is negative 36, giving us
x equals 10 plus or minus
the square root of negative
36, all divided by two.
The square root of negative 36 is equal
to the square root of
negative one times 36.
We know the square root of 36 is six.
The square root of negative one is i.
This is equal to 6i,
which gives us x equals 10
plus or minus 6i, all divided by two.
To simplify, let's break this up
us into two separate fractions,
or the real part and the imaginary part.
We would have 10 divided
by two, plus or minus
six halves i, so we have x
equals five plus or minus 3i.
So these are the two complex solutions.
If we're asked to list the solutions,
one solution is five plus 3i,
and the other is five minus 3i.
I hope you found this helpful.
