Video Transcript
Find the solution set of the
equation three 𝑥 squared plus four multiplied by 𝑥 plus one equals zero, giving
values in the set of real numbers to one decimal place.
In order to answer this question,
we firstly need to rearrange our equation so it is in the form 𝑎𝑥 squared plus
𝑏𝑥 plus 𝑐 equals zero. This will enable us to use the
quadratic formula to solve it. Distributing the parentheses by
multiplying four by 𝑥 and four by one gives us three 𝑥 squared plus four 𝑥 plus
four equals zero. Our values of 𝑎, 𝑏, and 𝑐 are
three, four, and four, respectively. The quadratic formula states that
𝑥 is equal to negative 𝑏 plus or minus the square root of 𝑏 squared minus four
𝑎𝑐 all divided by two 𝑎.
Substituting in our values, we have
𝑥 is equal to negative four plus or minus the square root of four squared minus
four multiplied by three multiplied by four all divided by two multiplied by
three. Four squared is 16, four multiplied
by three multiplied by four is 48, and two multiplied by three is six. As 16 minus 48 is negative 32, we
are left with 𝑥 is equal to negative four plus or minus the square root of negative
32 all divided by six.
At this stage, as we want solutions
to one decimal place, we would usually input our calculation into our
calculator. However, this calculation involves
taking the square root of a negative number, negative 32. And we know that square rooting a
negative number has no real solutions. And when we type it into the
calculator, we get a mathematical error. This means that there are no real
solutions to our equation. And the solution set of the
equation is the empty set. This leads us to a key fact about
the quadratic formula. If the expression underneath the
square root 𝑏 squared minus four 𝑎𝑐, known as the discriminant, is less than
zero, then there are no real solutions to our quadratic equation.
It is also worth considering what
this would look like graphically. The quadratic equation 𝑦 is equal
to three 𝑥 squared plus four multiplied by 𝑥 plus one is shown in the figure. We notice that the graph does not
intersect the 𝑥-axis. This confirms that there are no
real solutions to the equation. Any quadratic equation, where the
discriminant 𝑏 squared minus four 𝑎𝑐 is less than zero, will not intersect the
𝑥-axis.
