# Question Video: Determining Where the Graph of a Quadratic Equation Crosses the 𝑥-Axis by Factoring Mathematics

Factor fully the equation 𝑦 = 5𝑥² − 45. At which values of 𝑥 does the graph of 𝑦 = 5𝑥² − 45 cross the 𝑥-axis?

03:06

### 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.