We are asked to solve
the given quadratic inequality.
Looking at the inequality,
let's first set the
right side equal to zero
by subtracting eight on both sides,
the given inequality is the
equivalent to negative x squared
plus 3x minus eight greater
than or equal to zero.
The next step is to solve
the related equation,
which is the quadratic equation
negative x squared plus 3x
minus eight equals zero.
If we want to try to solve by factoring,
we need the x squared term to be positive,
so we can either multiply
both sides by negative one
or factor negative one from the left side,
let's factor negative
one from the left side,
which gives us negative
one times the quantity,
positive x squared minus
3x plus eight equals zero
and now see if the trinomial
inside the parentheses factors,
if it does factor, it will
factor into two binomial factors,
the first term is x squared,
which is equal to x times x,
the terms in the second
positions would need to be
the factors of positive
eight add to negative three,
but there are no factors of positive eight
to add a negative three,
which means this is not a factor
and therefore to solve
the quadratic equation,
we will need to use the quadratic
formula shown here below.
Let's go back up to this
form of the equation,
where a is equal to negative one,
b is equal to positive three
and c is equal to negative eight
and now perform substitution
into the quadratic formula,
the quadratic formula gives us x equals,
in the numerator, we have negative b,
which is negative three
plus or minus the square root
of b squared is three squared
and then minus 4ac is minus four
times negative one times negative eight,
all divided by two times a,
which is two times negative one,
now let's simplify,
we have negative three
plus or minus the square root
of three squared is nine,
then we have minus four times negative one
times negative eight, which is minus 32,
all divided by negative two,
nine minus 32 is negative 23
giving us x equals negative three
plus or minus the square
root of negative 23,
all divided by negative two,
notice how the radicand is negative
and therefore the square root
of negative 23 is imaginary,
remember the square root of
negative one is equal to i
and therefore this
simplifies to the quantity
negative three plus or minus i
square root 23 or square root 23i,
all divided by negative
two, so notice how here
the solutions to the equation are complex,
which means there are
no real solutions for x,
that satisfy the equation
or make the expression on
the left equal to zero,
which means all the real values for x
make the expression on the
left either greater than zero
or less than zero or positive or negative,
so normally we'd take the solutions
to the latter equation and
pop them on the number line
and then test the
intervals, but in this case,
we can select any real
number on the number line
and see if it satisfies the inequality,
if it satisfies the inequality,
then all real numbers will
satisfy the inequality,
if the value does not
satisfy the inequality,
then we have no solution
to the inequality.
So for our test value, again,
we can select any real number,
let's select the easiest number
to perform substitution with
of x equals zero
and let's use this form of the inequality,
rather than the original,
we substitute zero for x,
we have the opposite of zero squared
plus three times zero minus eight
greater than or equal to zero,
simplifying on the left, we
just have negative eight,
negative eight greater than
or equal to zero is false
and therefore all real
numbers would be false
and therefore there is no solution
to the quadratic inequality,
so we state no solution,
we can also denote no solution
using the empty setter null
set using this notation
or set notation with
no elements in the set.
Before we go, let's also verify
the solution graphically.
Considering this form of inequality,
to check this graphically,
we can graph the function,
f of x equals negative x
squared plus 3x minus eight
and because we want to determine
when this expression is
greater than or equal to zero,
we want to determine where the graph,
the function is above the x axis,
function values are
positive above the x axis,
equal to zero on the x axis
and negative below the x axis.
Looking at the graph, notice
all the function values
are below the x axis and
therefore they are negative
and negative values are never
greater than or equal to zero,
which verifies we have no solution
to the quadratic inequality.
I hope you found this helpful.
