Explainer: Solving Quadratic Equations Graphically

In this explainer, we will learn how to solve quadratic equations using function graphs.

Definition: Quadratic Equation

A quadratic equation is an equation of the form 𝑦=π‘Žπ‘₯2+𝑏π‘₯+𝑐, where π‘Ž, 𝑏, and 𝑐 are constants and π‘Žβ‰ 0.

It may also be written as 𝑦=π‘Ž(π‘₯βˆ’β„Ž)2+π‘˜, where π‘Ž is a constant and (β„Ž,π‘˜) is the vertex of the turning point.

When we are solving quadratic equations, we solve the equation to find the values of π‘₯ for which 𝑦=0. One of the ways we can solve a quadratic equation is by factoring, finding the values of π‘₯ such that the value of the factors is equal to zero. The solutions to a quadratic equation are also referred to as the roots of the equation. In this explainer, we will see how we can also use a graphical method to solve an equation. When we plot the graph of a function, the roots of the function will be the points at which the graph crosses the π‘₯-axis. At the points where the graph crosses the π‘₯-axis, the 𝑦-value will be zero. So, to find the roots of an equation 𝑦=𝑓(π‘₯), we can set 𝑓(π‘₯)=0.

Quadratic graphs have distinctive properties which can be used to help us identify points of interest about an equation. Whether inspecting a quadratic graph or using an equation to draw the graph, the following points are important to remember.

Characteristics of a Quadratic Graph

  • A quadratic equation in the form 𝑦=π‘Žπ‘₯2+𝑏π‘₯+𝑐 has a distinctive parabola shape. It will have a minimum vertex with the curve opening upward when the value of π‘Ž is greater than zero (π‘Ž>0). It will have a maximum vertex with the curve opening downward when the value of π‘Ž is less than zero (π‘Ž<0).
    Note that the value of π‘Ž cannot be equal to zero, as this would mean that there is no π‘₯2-term, and as such it would not be a quadratic equation.
  • The graph of a quadratic equation is symmetrical about the vertex.
  • The 𝑦-intercept, the coordinate where the parabola crosses the 𝑦-axis, of the function 𝑦=π‘Žπ‘₯2+𝑏π‘₯+𝑐 will always be at (0,𝑐).
  • The π‘₯-intercepts, where the curve crosses the π‘₯-axis, will be the points at which 𝑦=0. These are also called the roots of the equation. In a graph, we can identify these points by inspecting the graph.
  • It is useful to remember that a quadratic equation will have up to two roots. An equation with two roots will have a graph which crosses the π‘₯-axis twice, a repeated root will mean that the graph has a vertex on the π‘₯-axis, and an equation with no roots will have a graph that is above the π‘₯-axis.
  • In the graphs above, the first graph has two roots, the middle graph has one root, where the graph touches the π‘₯-axis, and the last graph has no roots.

Let us now look at some examples of how we can use the graph of a quadratic equation to solve it.

Example 1: Solving a Quadratic Equation Graphically

The graph shows the curve with the equation 𝑦=𝑓(π‘₯). What is the solution set of the equation 𝑓(π‘₯)=0?

Answer

The solution set of the equation 𝑓(π‘₯)=0 is the values of π‘₯ at which the curve crosses the π‘₯-axis. From the graph, it is evident that the curve crosses the π‘₯-axis when π‘₯=βˆ’2 and π‘₯=2.

Therefore, the solution set is π‘₯={βˆ’2,2}.

Example 2: Finding the Roots of a Quadratic Equation Using a Graph

The roots of a quadratic equation can be read from the graph. What are they?

Answer

The roots of the quadratic equation are the values of π‘₯ at which the curve crosses the π‘₯-axis. Inspecting the graph, we can see that the function cuts the π‘₯-axis at π‘₯=βˆ’12 and π‘₯=βˆ’32.

Therefore, the roots of the equation are βˆ’12 and βˆ’32.

Example 3: Solving a Quadratic Equation Graphically

The graph below shows the function 𝑓(π‘₯)=π‘₯2βˆ’2π‘₯+3. What is the solution set of 𝑓(π‘₯)=0?

Answer

The solution set of the equation 𝑓(π‘₯)=0 is the values of π‘₯ at which the curve crosses the π‘₯-axis.

In this graph, it is evident that the graph does not cross the π‘₯-axis. Therefore, there are no values of π‘₯ for which 𝑓(π‘₯)=0, and the solution set is the empty set βˆ….

We will now look at an example of how we can draw our own graph to solve an equation. Here, we will use a table of values to find a number of coordinates to plot and then join them to make a curve. It is useful to remember that using a table of values will allow us to plot the graph of any polynomial equation.

Example 4: Solving a Quadratic Equation Graphically

By drawing a graph of the function 𝑓(π‘₯)=2π‘₯2βˆ’3π‘₯, find the solution set of 𝑓(π‘₯)=0.

Answer

We can begin drawing a graph of the function by finding coordinates on the curve. We can set up a table of values and calculate 𝑓(π‘₯) for the selected π‘₯-values.

π‘₯βˆ’2βˆ’10123
2π‘₯28202818
3π‘₯βˆ’6βˆ’30369
𝑓(π‘₯)=2π‘₯2βˆ’3π‘₯1450βˆ’129

We can plot the points (βˆ’2,14), (βˆ’1,5), (0,0), (1,βˆ’1), (2,2), and (3,9) and join them with a smooth curve.

The solution set of 𝑓(π‘₯)=0 is the values of π‘₯ at which the curve crosses the π‘₯-axis. We can see that the graph crosses the π‘₯-axis at π‘₯=0 and π‘₯=32. Therefore, the solution set is 0,32.

We will now look at an example where we use the solution of the equation, the roots, to help us draw a graph. This can be a quicker method of sketching a graph than using a table of values; however, we need to identify other properties of the equation in order to have more than two coordinates with which to draw the graph.

Example 5: Drawing a Quadratic Graph Using the Roots of the Equation

  1. Solve π‘₯2βˆ’10π‘₯+25=0 by factoring.
  2. Draw the graph of 𝑓(π‘₯)=π‘₯2βˆ’10π‘₯+25.

Answer

Part 1

We can factor the equation π‘₯2βˆ’10π‘₯+25=0 as (π‘₯βˆ’5)(π‘₯βˆ’5)=0.

To solve this, recall that when we have two numbers, 𝐴 and 𝐡, giving 𝐴𝐡=0, this means that 𝐴=0 or 𝐡=0. Here, we have (π‘₯βˆ’5)=0or(π‘₯βˆ’5)=0.

So, π‘₯=5orπ‘₯=5.

As this is a repeated root, we can say that there is just one solution: π‘₯=5.

Part 2

To draw the graph of the function, we can start with the roots of the equation. Here, we have a repeated root of π‘₯=5. When we have a repeated root, this is the vertex of the equation. Therefore, the vertex here is at (5,0).

Recall that a function 𝑓(π‘₯)=π‘Žπ‘₯2+𝑏π‘₯+𝑐, where π‘Ž, 𝑏, and 𝑐 are constants, has a 𝑦-intercept at (0,𝑐). This means that the 𝑦-intercept of 𝑓(π‘₯)=π‘₯2βˆ’10π‘₯+25 is at (0,25).

We can recall that a quadratic equation with a positive coefficient of π‘₯2 has a minimum vertex, with the parabola curve opening upward. Since the graph of a quadratic equation is symmetrical about the vertex, we can say that the coordinate (10,25) will also lie on the curve.

We can plot the points (5,0), (0,25), and (10,25) and join them with a smooth curve to complete the question.

When solving an equation using a graph, we have been using the intersection of the quadratic equation and the π‘₯-axis to find the π‘₯-values when 𝑦=0. We will now look at an example where we can use the intersection of a quadratic curve and another line to solve a problem.

Example 6: Solving a Quadratic Equation Graphically

The graph shows the function 𝑓(π‘₯)=2π‘₯2βˆ’4π‘₯βˆ’6.

  1. What is the solution set of 𝑓(π‘₯)=0?
  2. What is the solution set of 𝑓(π‘₯)=βˆ’6?

Answer

Part 1

The solution set of 𝑓(π‘₯)=0 is the values of π‘₯ where the curve crosses the π‘₯-axis. We can see from the graph that it crosses the π‘₯-axis at π‘₯=βˆ’1 and π‘₯=3. Therefore, the solution set of 𝑓(π‘₯)=0 is π‘₯={βˆ’1,3}.

Part 2

The solution set of 𝑓(π‘₯)=βˆ’6 is the values where the curve crosses the line 𝑓(π‘₯)=βˆ’6. The line 𝑓(π‘₯)=βˆ’6 is a horizontal line crossing through all the points where the 𝑦 coordinate is βˆ’6. We can add this line to the graph and determine the coordinates where the two functions 𝑓(π‘₯)=2π‘₯2βˆ’4π‘₯βˆ’6 and 𝑓(π‘₯)=βˆ’6 intersect.

We can identify from the graph that the points of intersection are at π‘₯=0 and π‘₯=2. Therefore, the solution set of 𝑓(π‘₯)=βˆ’6 is {0,2}.

Key Points

  • We can solve a quadratic equation in a number of ways, either using an algebraic method, such as factoring, or graphically, by inspecting the graph of the function. The benefit of an algebraic solution is that it will result in an exact answer. A graphical method is an approximate solution; therefore, it is often more difficult to use when there are noninteger root values. However, it can be an easier and faster method of solving a quadratic equation and allows us to visually compare a number of quadratic equations.
  • To solve a quadratic graphically, we first draw the graph of a function by creating a table of values. We can then inspect the graph to find the points where the graph crosses the π‘₯-axis.
  • It is important to remember that we may have two, one, or no solutions to a quadratic depending on whether the function crosses the π‘₯-axis twice, once, or never.
  • Using the features of the graph of a function can be a useful method to solve equations when using graphing software such as a graphical calculator.

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