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.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.