In this video, we will learn how to solve quadratic equations using function graphs. First, we remember that quadratic equations could be in the form 𝑦 equals 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐, where 𝑎, 𝑏, and 𝑐 are constants and 𝑎 cannot equal zero. This is the standard form for quadratic equations. We might also see them as 𝑦 equals 𝑎 times 𝑥 minus ℎ squared plus 𝑘, where 𝑎 is a constant and ℎ, 𝑘 is the vertex or the turning point of the function. And we call this the vertex form. But we’re not just looking at quadratic equations. We’re looking at how to solve them.
Solving an equation is to find the values of 𝑥 for which 𝑦 is zero. One of the ways we solve quadratic equations is by factoring. For example, if we wanted to solve 𝑦 equals 𝑥 squared plus four 𝑥 plus three by factoring, we set 𝑦 equal to zero. And then we take out the terms 𝑥 plus one and 𝑥 plus three. We set both factors equal to zero. And then we get the values for 𝑥 which makes 𝑦 equal to zero in this equation. And we call these roots or solutions. But in this video, we want to solve by graphing. We’ll do this by identifying points of interest on the graph. To do that, we’ll consider the characteristics of quadratic graphs.
Some important characteristics of a quadratic graph. Quadratic-function graphs have a distinctive parabola shape. We generally think of it as a U shape pointing upward or downward. They will open upward when 𝑎 is greater than zero, when 𝑎 is positive. Remember that 𝑎-value is the coefficient of the 𝑥 squared term when we write our quadratic equation in standard form. And likewise, the graph will open downward when 𝑎 is less than zero, when 𝑎 is negative. We can also say that when 𝑎 is positive, there will be a minimum value on the graph and that when 𝑎 is negative, there will be a maximum value on the graph.
And we remember that 𝑎 cannot be equal to zero as that would mean the graph would not be quadratic but linear. We also know that quadratic-function graphs are symmetrical about the vertex. We could fold the graph in half at the vertex. And both sides would lie on top of each other. The quadratic-function graphs have a 𝑦-intercept at the point zero, 𝑐. The graphs will have 𝑥-intercepts or 𝑥-intercept where 𝑦 equals zero. Remember, these are also called roots or solutions. The graph could have two solutions, one solution, or no solutions if the graph does not cross the 𝑥-axis at all.
To solve a quadratic function by graphing, we will be trying to identify the places where the graph crosses the 𝑥-axis. Let’s try to do that now.
The diagram shows the graph of 𝑦 equals 𝑓 of 𝑥. What is the solution set of the equation 𝑓 of 𝑥 equals zero?
We know that 𝑦 equals 𝑓 of 𝑥. And we’re looking for the place where 𝑓 of 𝑥 equals zero. That will be the place for this quadratic where 𝑦 equals zero. And 𝑦 equals zero at the 𝑥-intercept. 𝑦 equals zero along the 𝑥-axis. And this function is crossing the 𝑥-axis at negative two. Therefore, the 𝑥-intercept is negative two, zero. But the solution set will be the 𝑥-value that makes 𝑓 of 𝑥 equal to zero. And that means instead of coordinate form, we’re only interested in the 𝑥-coordinate. This function equals zero when 𝑥 equals negative two. Therefore, the solution set will only include the value negative two.
Here’s another example.
Consider the graph. The roots of the quadratic can be read from the graph. What are they?
We should immediately be thinking about what the roots of a graph are. The roots of a quadratic equation are the 𝑥-values that make 𝑦 equal to zero. And that means we need to look at the places this graph crosses the 𝑥-axis. The 𝑥-axis is the line 𝑦 equals zero. And here, we see the two places our graph is crossing.
Between zero and negative one, there are five spaces. And our point falls in the middle of the third space. It falls halfway between zero and negative one. That first location is then 𝑥 equals negative one-half. When 𝑥 equals negative one-half, 𝑦 equals zero. And the second root is halfway between negative one and negative two, which is negative one and a half. But for consistency, we could write it as negative three-halves. And so we can say that the roots of this graph are negative one-half and negative three-halves.
Here’s a third example.
The graph shows the function 𝑓 of 𝑥 equals 𝑥 squared minus two 𝑥 plus three. What is the solution set of 𝑓 of 𝑥 equals zero?
If we’re looking for 𝑓 of 𝑥 equal to zero, we’re looking for the 𝑥-values that make 𝑥 squared minus two 𝑥 plus three equal to zero. And on the graph, that will be the place where 𝑦 equals zero. And we know that 𝑦 equals zero along the 𝑥-axis. In order for this function to have a solution of 𝑓 of 𝑥 equals zero, it would need to cross the 𝑥-axis. This function does not cross the 𝑥-axis. Therefore, it has zero solutions or zero roots. And therefore, the solution set here is the empty set; there is no solution.
In our next example, we’ll draw our own graph to solve an equation.
By drawing a graph of the function 𝑓 of 𝑥 equals two 𝑥 squared minus three 𝑥, find the solution set of 𝑓 of 𝑥 equals zero.
Our equation is 𝑓 of 𝑥 equals two 𝑥 squared minus three 𝑥. Before we start drawing our graph, it’s helpful to have a few coordinates so that we can sketch the curve. We can use the tables to do this. If we calculate the 𝑥-values, negative two, negative one, zero, one, two, that should give us some idea of the shape of this graph. Our first box would be two times negative two squared minus three times negative two, which equals 14. Next, we have two times negative one squared minus three times negative one, which is five. When we plug in zero, the result is zero. When we plug in one, we get negative one. When we plug in two, we get two.
The top values, the 𝑥-values, represent the domain of the function, what we can plug in for 𝑥. And the bottom values represent the range, what we’ll need for the 𝑦-values. In these values, we have a range that’s lowest point is negative one and highest point is 14. Now, these are just from the points we’ve chosen. That doesn’t mean it’s the range of the full function. But it does give us an idea of what the scale of the 𝑥- and 𝑦-axis should be. We can graph a point at negative two, 14; negative one, five; zero, zero; one, negative one; and two, two.
When we’re looking at this, we see that it might be helpful to have an additional point on the 𝑥-axis, so we could solve for three. When we plug three into the equation, we get nine as the output. And that just gives us one more point to be able to sketch this graph. It won’t be perfect, but you can try to get a smooth curve between the points. And when we do that, since we’re looking for the solution set of 𝑓 of 𝑥 equals zero, we’re looking for the 𝑥-intercepts. We’re looking for the places where this function crosses the 𝑥-axis. The first one is very clear. It’s the point zero, zero. And the second intersection happens halfway between one and two. We have a solution when 𝑥 equals zero and a solution when 𝑥 is one and a half which makes our solution set zero and three-halves or zero and one and a half.
Remember, this is not a coordinate point. These are two different 𝑥-values which when inputted into the function, the output is zero.
Now, we’re going to look at an example where we’ll use the solutions of the equation, the roots, to help us draw a graph. This can be a quicker method than using a table of values every time. However, we’ll still need to identify some other coordinates to help us draw a graph.
(1) Solve 𝑥 squared minus 10𝑥 plus 25 equals zero by factoring. (2) Draw the graph of 𝑓 of 𝑥 equals 𝑥 squared minus 10𝑥 plus 25.
For the first part of this problem, we’ll need to factor 𝑥 squared minus 10𝑥 plus 25. Since 𝑥 squared has a coefficient of one, we know that each of the factors will have one 𝑥 term. And then we need to consider two values that when multiplied together equal positive 25, but when added together equal negative 10. We have one and 25; negative one, negative 25; positive five, positive five; and negative five, negative five.
When we add negative five plus negative five, we get negative 10. And so we found the pairs. And we realize that this is 𝑥 minus five times 𝑥 minus five. Or we could say it’s 𝑥 minus five squared. But again We need to solve for 𝑥 here. And that means we should get 𝑥 by itself, take the square root of both sides, and we get 𝑥 minus five equals zero. And so we can say that the solution or the root of this equation is 𝑥 equal to five. The equation only has one root. And that means it will only cross the 𝑥-axis one time. This is all we need for the first part of the question.
But in order to draw this graph, we’ll need a little bit more information. We know we have the root five, zero. And because there is only one root, because this is a repeated root, the point five, zero will also be the vertex of the equation. Our equation was originally given to us in standard form 𝑦 equals 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐. And when the function is given in standard form, the 𝑦-intercept is located at zero, 𝑐. For us, that constant value, that 𝑐-value, is 25. And we can say that we have a 𝑦-intercept at zero, 25.
We can use these values to go ahead and sketch our 𝑥- and 𝑦-axis. Because the coefficient of our 𝑥 squared, our 𝑎-value, is positive one, we can say that the parabola will be opening upward and we can say that the vertex will be a minimum. Since five, zero is the minimum value, we can say that zero is the lowest value for our 𝑦-coordinates. Then we’ll add the scale to our graph using this information. We have a minimum at five, zero, a 𝑦-intercept at zero, 25. And because we know that parabolas are symmetrical about the vertex, the 25 is five units away from the 𝑥-coordinate of the vertex. And that means five coordinates to the right of the vertex would also be equal to 25. There would be a coordinate at 10, 25 as well as at zero, 25. And from there, we try and use a smooth curve to connect the points, which completes our graph.
In all of our previous examples, we’ve been solving for the 𝑥-values when 𝑦 equals zero. We’ll now look at an example where we can use the intersection of a quadratic curve and another line.
The graph shows the function 𝑓 of 𝑥 equals two 𝑥 squared minus four 𝑥 minus six. What is the solution set of 𝑓 of 𝑥 equals zero? What is the solution set of 𝑓 of 𝑥 equals negative six?
The solution set for 𝑓 of 𝑥 equals zero will be the places where two 𝑥 squared minus four 𝑥 minus six equals zero. And on the graph, that’s the place where they cross the 𝑥-axis. The 𝑦 equals zero across the 𝑥-axis. In this case, we have a solution at negative one and at positive three. So for 𝑓 of 𝑥 equal to zero, the solution set is negative one, three. Remember, this is not a coordinate. These are the 𝑥-values for which 𝑓 of 𝑥 equals zero. And we’ve given them in set notation.
The 𝑥-axis, as we’ve said, is the place where 𝑦 equals zero. The line 𝑓 of 𝑥 equals zero is the 𝑥-axis. But what would we do if we were using a graph to look at 𝑓 of 𝑥 equals negative six? This would be the place or the places where our quadratic cross the line 𝑦 equals negative six. And in this graph, we see that’s happening twice. It happens when 𝑥 equals zero and when 𝑥 equals two. In set notation, we’re saying the two 𝑥-values for which 𝑓 of 𝑥 equals negative six are zero and two.
Let’s review the key points when dealing with solving quadratic equations graphically. Solving quadratic equations graphically works by inspecting the graph of a function. This process yields an approximate solution. Therefore, when dealing with noninteger roots, it may be better to solve for the roots algebraically. We remember that quadratic equations can have one, two, or zero roots. And when we look at this on the graph, that means it can cross the 𝑥-axis one time, two times, or not at all.
And finally, to solve a quadratic equation graphically, we find the place on the graph where the equation crosses the 𝑥-axis. It might be worth noting here that this can be a useful method if you’re using a graphing calculator to solve this kind of problem. You can enter quadratic equations into your graphing calculator. And it will produce a graph for you to use. Even when working with a calculator, the procedure for identifying the 𝑥-intercepts will remain the same, looking for the places where the equation crosses the 𝑥-axis.