Video: Determining Where the Graph of a Quadratic Equation Crosses the π‘₯-Axis

At which values of π‘₯ does the graph of 𝑦 = 4π‘₯Β² βˆ’ 20 cross the π‘₯-axis?

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Video Transcript

At which values of π‘₯ does the graph of 𝑦 equals four π‘₯ squared minus 20 cross the π‘₯-axis?

Remember, we can actually describe the π‘₯-axis with an equation. It has the equation 𝑦 equals zero. And so, to find the values of π‘₯ for which the graph crosses the π‘₯-axis, we set 𝑦 equal to zero and solve for π‘₯. This is sometimes called finding the roots of the equation. We need to solve the equation four π‘₯ 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 π‘₯ 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 π‘₯ squared minus five plus five, which is π‘₯ squared plus zero or simply π‘₯ 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 π‘₯ is equal to plus or minus root five. And so, we have two solutions to the equation four π‘₯ squared minus 20 equals zero. They are π‘₯ is equal to the square root of five and π‘₯ is equal to negative square root of five.

Before we assume that these are the values of π‘₯ for which the graph of 𝑦 equals four π‘₯ squared minus 20 does cross the π‘₯-axis, let’s check that what we’ve done is correct. We do this by substituting π‘₯ equals root five and π‘₯ equals negative root five into the expression four π‘₯ squared minus 20. We want to see that we do indeed get zero. So, when π‘₯ 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 π‘₯ 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 π‘₯ for which the graph of 𝑦 equals four π‘₯ squared minus 20 crosses the π‘₯-axis are π‘₯ equals root five and π‘₯ equals negative root five.

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