Find the solutions to two 𝑥
squared minus six 𝑥 plus three equals zero. Give your answer to two decimal
The first thing we notice about
this question is that the question says to give our answer to two decimal
places. This tells us that our solutions
are not going to be whole numbers. And so we will not be able to
factorize this equation. Therefore, we’re going to need to
use the quadratic formula. Let’s remind ourselves what the
quadratic formula is.
If we have an equation of the form
𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero, then the quadratic formula tells us that
the solution to this equation will be 𝑥 equals minus 𝑏 plus or minus the square
root of 𝑏 squared minus four 𝑎𝑐 all over two 𝑎. Now, let’s compare the equation
𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero to the equation we’re given in the
question. Comparing the coefficients, we can
see that 𝑎 equals two, 𝑏 equals negative six, and 𝑐 equals three.
Now, all we need to do to find our
solutions is to substitute these values into the quadratic formula. It can certainly be helpful to put
a bracket around the negative number so that way we don’t get confused when squaring
it or multiplying it by other numbers. What we are left with is 𝑥 is
equal to minus negative six plus or minus the square root of negative six squared
minus four times two times three all over two times two.
Now, we can simplify this. We must be careful to remember that
a negative number timesed by another negative number gives us a positive number. And likewise, if we’re subtracting
a negative number, then we also get a positive number. So we start with minus negative six
giving us positive six. Then, we have plus or minus the
square root of negative six squared. So that’s minus six times minus six
giving us positive 36 minus four times two times three which is 24 all over two
times two which is just four. Next, we can simplify the square
root to give us 𝑥 is equal to six plus or minus the square root of 12 all over
From this, we obtain two solutions:
one where we take the plus or minus as a plus and the other where you take the plus
or minus as a minus. So our two solutions are 𝑥 is
equal to six plus the square root of 12 over four or 𝑥 is equal to six minus the
square root of 12 over four.
Using our calculator, we see that
these are the two solutions to the equation. However, the question asked us to
give our answer to two decimal places. So now, we need to round our
answers. Rounding to two decimal places, we
get solutions of 𝑥 is equal to 2.37 or 𝑥 is equal to 0.63.