At which values of 𝑥 does the
graph of 𝑦 equals four 𝑥 squared minus 20 cross the 𝑥-axis?
Remember, we can actually describe
the 𝑥-axis with an equation. It has the equation 𝑦 equals
zero. And so, to find the values of 𝑥
for which the graph crosses the 𝑥-axis, we set 𝑦 equal to zero and solve for
𝑥. This is sometimes called finding
the roots of the equation. We need to solve the equation four
𝑥 squared minus 20 equals zero. So, how do we do this? Well, specifically with quadratic
equations, it’s often useful to spot if there are any common factors. Well, here, both four and 20 share
the factor of four, so we divide through by four. When we do, we find that 𝑥 squared
minus five is equal to zero.
Next, we want to add five to both
sides of this equation. When we do, on the left-hand side
we get 𝑥 squared minus five plus five, which is 𝑥 squared plus zero or simply 𝑥
squared. And then, on the right, we simply
have five. The final thing we want to achieve
is to get rid of somehow this square. And so, we do the opposite to
squaring. We square root both sides of the
equation. Remember, though, we need to find
both the positive and negative square root of five. So, we find 𝑥 is equal to plus or
minus root five. And so, we have two solutions to
the equation four 𝑥 squared minus 20 equals zero. They are 𝑥 is equal to the square
root of five and 𝑥 is equal to negative square root of five.
Before we assume that these are the
values of 𝑥 for which the graph of 𝑦 equals four 𝑥 squared minus 20 does cross
the 𝑥-axis, let’s check that what we’ve done is correct. We do this by substituting 𝑥
equals root five and 𝑥 equals negative root five into the expression four 𝑥
squared minus 20. We want to see that we do indeed
get zero. So, when 𝑥 is equal to root five,
we get four times root five squared minus 20. Root five squared is five and four
times five is 20. So, we have 20 minus 20 which is
zero, as required.
Let’s try this with 𝑥 equals
negative root five. We get four times negative root
five squared minus 20. Well, actually, negative root five
squared is five again, so we do indeed get zero, as required. And this means the roots of our
equation or the values of 𝑥 for which the graph of 𝑦 equals four 𝑥 squared minus
20 crosses the 𝑥-axis are 𝑥 equals root five and 𝑥 equals negative root five.