Video Transcript
Factor fully the equation 𝑦 is
equal to five 𝑥 squared minus 45. At which values of 𝑥 does the
graph of 𝑦 equals five 𝑥 squared minus 45 cross the 𝑥-axis?
In order to factor our equation, we
firstly look for any common factors. The highest common factor of five
𝑥 squared and negative 45 is five. This means that we can rewrite our
equation as 𝑦 is equal to five multiplied by 𝑥 squared minus nine as five
multiplied by 𝑥 squared is five 𝑥 squared and five multiplied by negative nine is
negative 45. The expression inside the
parentheses is written in the form 𝑥 squared minus 𝑎 squared.
This can be factored using the
difference of two squares. 𝑥 squared minus 𝑎 squared is
equal to 𝑥 plus 𝑎 multiplied by 𝑥 minus 𝑎. As the square root of nine is equal
to three, 𝑥 squared minus nine can be rewritten as 𝑥 plus three multiplied by 𝑥
minus three. Our equation simplifies to 𝑦 is
equal to five multiplied by 𝑥 plus three multiplied by 𝑥 minus three.
In the second part of the question,
we need to find the values of 𝑥 where the graph crosses the 𝑥-axis. We know this occurs when 𝑦 is
equal to zero. Any quadratic equation where the
coefficient of 𝑥 squared is positive will be U-shaped. In this question, the equation 𝑦
is equal to five 𝑥 squared minus 45 crosses the 𝑥-axis at two points. These occur when five 𝑥 squared
minus 45 is equal to zero.
Using our answer to the first part
of the question, we see that five multiplied by 𝑥 plus three multiplied by 𝑥 minus
three must be equal to zero. Dividing both sides of this
equation by five, we have 𝑥 plus three multiplied by 𝑥 minus three is equal to
zero. For this to be true, one of our
parentheses must equal zero, either 𝑥 plus three equals zero or 𝑥 minus three is
equal to zero. For the first equation, we subtract
three from both sides, such that 𝑥 is equal to negative three. We can solve our second equation by
adding three to both sides such that 𝑥 is equal to three.
The two values of 𝑥 for which the
graph crosses the 𝑥-axis are negative three and three. These are the two answers to the
second part of the question.
