Solve for 𝑥: root three 𝑥 squared
minus two root two 𝑥 minus two root three is equal to zero.
Now, this equation is a quadratic
equation as the highest power of 𝑥 is a two. We have this 𝑥 squared term
here. We have a choice of methods for how
we can solve a quadratic equation: we could try to factorise it, we could use the
quadratic formula, or we could complete the square.
In this quadratic equation, the
coefficients are not particularly straightforward. They’re given in terms of surds,
which means it’s probably not going to factorize easily. Our best option then is to apply
the quadratic formula. The quadratic formula tells us that
the roots of the general quadratic equation 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 is equal
to zero are given by 𝑥 is equal to negative 𝑏 plus or minus the square root of 𝑏
squared minus four 𝑎𝑐 all over two 𝑎.
Here, 𝑎 is the coefficient of 𝑥
squared, 𝑏 is the coefficient of 𝑥, and 𝑐 is the constant term. So for this quadratic equation, 𝑎
is equal to root three, 𝑏 is equal to negative two root two, and 𝑐 is equal to
negative two root three. So we can substitute the values of
𝑎, 𝑏, and 𝑐 into our quadratic formula. And it gives two root two first of
all because this is negative negative two root two plus or minus the square root of
negative two root two squared — that’s 𝑏 squared — minus four multiplied by root
three multiplied by negative two root three — that’s four 𝑎𝑐 — all over two 𝑎
which is two root three.
Now, we need to simplify this. And we can start underneath the
square root first of all, where we have negative two root two squared. Now, this is equal to negative two
root two multiplied by negative two root two. And we can do these multiplications
in any order. So if we multiply negative two by
negative two first, this gives four and root two multiplied by root two gives
two. So we have four multiplied by two
which is equal to eight.
Also within the square root, we
have that bracket which is four multiplied by root three multiplied by negative two
root three. So using the same idea, we can
multiply the four by the negative two, which gives negative eight and then root
three by root three which gives three. So we have negative eight
multiplied by three which is negative 24.
Within the square root then, we
have eight minus negative 24 and those two negative signs together make a
positive. So we have eight plus 24, which is
equal to 32. So the value within that square
root is actually simplified very nicely. And we’re left with 𝑥 is equal to
two root two plus or minus root 32 all over two root three.
Next, we want to see if we can
simplify this root 32 any further. And to simplify a surd or a square
root, we want to look for the square factors of the number underneath the square
root. 32 is of course equal to 16
multiplied by two. And 16 is a square number. So we can express the square root
of 32 as the square root of 16 multiplied by the square root of two.
Now, remember 16 is a square
number. It’s equal to four squared. So the square root of 16 is just
four. And therefore, root 32 simplifies
to four root two. So the value of our root simplifies
to two root two plus or minus four root two all over two root three. Now, we can actually cancel a
factor of two from every term in this expression. And it leaves us with root two plus
or minus two root two all over root three.
So our two unsimplified roots then,
firstly, we have root two plus two root two all over root three and secondly, we
have root two minus two root two all over root three. Now, the numerators of each of
these roots can be simplified. For the first root, we had root two
plus two root two which gives three root two and for the second root we had root two
minus two root two which gives negative root two.
So our two roots become three root
two over root three and negative root two over root three. Now, both of these roots currently
have a surd in the denominator. And so we need to rationalize
this. To rationalize, we need to multiply
by root three over root three. Now, this doesn’t change the value
of either root as this fraction is equivalent to one.
For the first root, we have three
root two multiplied by root three in the numerator, which gives three root six, and
root three multiplied by root three in the denominator which gives three. And now, we’ve rationalized the
denominator. We can, however, simplify this root
slightly further as the factor of three in the numerator can cancel with the factor
of three in the denominator. And we’re just left with root
six. This can’t be cancelled any further
as six doesn’t have any square factors.
For the second root, we have
negative root two multiplied by root three in the numerator, which is equal to
negative root six, and in the denominator root three multiplied by root three which
is equal to three. Now, this can’t be simplified any
further. But it now has a rational
So our solution to the given
quadratic equation is 𝑥 is equal to root six or negative root six over three.