Video Transcript
At which values of 𝑥 does the graph of the equation 𝑦 is equal to negative one times 𝑥 minus three multiplied by 𝑥 plus eight cross the 𝑥-axis?
In this question, we’re given a fully factored quadratic equation. And we need to determine at which values of 𝑥 does the graph of this quadratic equation cross the 𝑥-axis. To answer this question, we’ll start by recalling what it means to graph an equation.
If a point with coordinates 𝑥, 𝑦 lies on our curve, it must be a solution to the equation. In other words, if the point with coordinates 𝑎, 𝑏 lies on our curve, then when 𝑥 is equal to 𝑎 and 𝑦 is equal to 𝑏, the equation must hold true. Both sides of the equation will be equal. This then allows us to determine what it means for the curve to cross the 𝑥-axis. Every point on the 𝑥-axis will have a 𝑦-coordinate of zero. In other words, these will be the points on our curve where 𝑦 is equal to zero. And since every point on the curve satisfies the equation, we can solve for these values by substituting 𝑦 is equal to zero into the equation. This gives us that zero is equal to negative one times 𝑥 minus three multiplied by 𝑥 plus eight. Any value of 𝑥 which solves this equation is the 𝑥-coordinate of a point on the curve which crosses the 𝑥-axis.
And now we can see we have a product of three numbers which is equal to zero. And we can recall for the product of numbers to be equal to zero, one of the factors must be equal to zero. The first factor is equal to negative one. It doesn’t matter what value of 𝑥 we choose. This will never be equal to zero. If our second factor is equal to zero, we get that 𝑥 minus three is equal to zero. And we can solve for 𝑥 by adding three to both sides of the equation. We get that 𝑥 is equal to three.
We can do the same for our third factor. We want to solve 𝑥 minus eight is equal to zero. We can subtract eight from both sides of the equation to get that 𝑥 will be equal to negative eight. Therefore, if 𝑥 is equal to three, our second factor is equal to zero. So both sides of the equation are equal to zero, which means the point with coordinates three, zero lies on the graph of our equation. And this is a point on the 𝑥-axis. And the same is true for the point with coordinates negative eight, zero. It solves the equation, so it lies on the graph of the equation. And it’s on the 𝑥-axis. And since these are the only solutions to this equation, these are the only points where the graph crosses the 𝑥-axis.
Therefore, we were able to find all of the values of 𝑥 where the graph of the equation 𝑦 is equal to negative one times 𝑥 minus three multiplied by 𝑥 plus eight crosses the 𝑥-axis. It’s when 𝑥 is equal to three and when 𝑥 is equal to negative eight.
