Solving quadratic equations by Graphing

Solving quadratic equations by Graphing | Free Math Class

In the series of classes on solving quadratic equations, this time we are writing on the graphical method and once again we are assuming that by now you have a strong grip on solving quadratic equations but the factor method and using completing the square technique.

In case you are interested in a quick recap then you can read our previous blogs on Factoring a quadratic equation and solving by completing the square method. These blogs will add more insights to your current knowledge.

By now you know and are getting eager to understand another way of solving a quadratic equation that is by solving it graphically. Isn’t it? Here you will be working with the roots of a quadratic equation and treating them as the x-intercepts of the graph to find the solutions to a corresponding equation. Just a suggestion before you begin understanding the solutions, put yourself in a laser mode and we assure you that once you read the complete blog, you will be clear with this topic in depth. 


So, let’s start on how to use the graph of a quadratic function to solve a quadratic equation

It is important to know that although graphing is a popular method of solving quadratic equations, as compared to the other two methods, the graphical method gives an estimate to the solution. 

You will arrive at different scenarios depending on where and how many points the quadratic function crosses the x-axis on the graph. For example, if the function crosses the graph at two points then you will have two solutions or two roots. If the graph touches the x-axis at a single point, then you will get one solution or one root only. If the graph does not intersect with the x-axis at all then the equation has no real solution. 

Below our math experts show the three types of solutions that a quadratic equation can have by taking a step-by-step approach: two solutions, one solution and no real solution.


Graph Quadratic Equation

The standard form of a quadratic equation is ax^2+bx+c =  0. 

The shape of the graph depends upon the coefficients a, b, and c. 

If a > 0, the graph will open upward, and if  a < 0, the graph will open downward. Also, a larger positive value of the coefficient of x^2 means the graph will increase faster and it will be thinner, vice versa is also true. 

And, the axis of symmetry can be found by x=(-b)/2a. The vertex of the parabola will also lie on the axis of the symmetry. 

Finally, we will use the coefficient c, which controls the height of the parabola. The point (0,c) describes the y-intercept of the parabola. 

We hope this is clear to you. Now, let’s have a look at these solutions! 


Graph Quadratic Equation with Two Solution

Example 1: x^2 -4 = 0

Explanation: After comparing the given equation with the standard form of quadratic equation ax^2+bx+c = 0, we have a=1,b=0, and c= – 4.

We have a > 0. So the graph will open upward. 

Now, let us find the axis of symmetry:

x = (-b)/2a = (-0)/(2×1) = 0

So, the axis of symmetry would be at x = 0, which is our y-axis.

Let us identify the vertex of the parabola. Plugging x=(-b)/2a in the equation, will give us the y-coordinate of the vertex. 

We know, x = (-b)/2a = (-0)/(2×1) = 0

f(x) = x^2 – 4

f(0) = 0^2 – 4 = – 4

Hence, the vertex of the parabola will be at (0, -4).

Now, let us find the y-intercept of the parabola. 

The point (0,c)will be the y-intercept of the parabola. So, (0,-4).

Now, let us use the information to plot the graph.

Vertex of the parabola will be at (0,-4). And, the axis of symmetry would be at x = 0.

 

Graph Quadratic Equation with Two Solution

We can see the graph of the function crosses the x-axis at two points, so it will have two solutions.

Graph Quadratic Equation with One Solution

Example 1: x^2 – 6x +9 = 0

Explanation: After comparing the given equation with the standard form of quadratic equation ax^2+bx+c = 0, we have a=1,b=-6, and c=9.

We have a > 0. So the graph will open upward. 

Now, let us find the axis of symmetry:

x=(-b)/2a=(-(-6))/(2×1)=6/2=3

So, the axis of symmetry would be at x=3.

Let us identify the y-coordinate of the vertex of the parabola by plugging x=(-b)/2a in the equation.

We know, x=(-b)/2a=(-(-6))/(2×1)=6/2=3.

f(x)=x^2 -6x +9

f(3)=3^2-6(3)+9=9 -18 +9= 0

Hence, the vertex of the parabola will be at (3,0).

The point (0,c) will be the y-intercept of the parabola. So, (0,9).

Now, let us use the information to plot the graph.

Vertex of the parabola will be at (3,0). And, the axis of symmetry would be at x=3. And y-intercept of the parabola would be at (0,9).

Graph Quadratic Equation with One Solution

The graph of the function crosses the x-axis at one point, so it will have one solution.


Graph Quadratic Equation with No Real Solution

Example 1: x^2-10x+24=0

Explanation: After comparing the given equation with the standard form of quadratic equation ax^2+bx+c=0, we have a=1,b=-10, and c=24.

We have a > 0. So the graph will open upward. 

Now, let us find the axis of symmetry:

x=(-b)/2a=(-(-10))/(2×1)=10/2=5

So, the axis of symmetry would be at x=5.

Let us identify the y-coordinate of the vertex of the parabola by plugging x=(-b)/2a in the equation. 

We know,x=(-b)/2a=(-(-10))/(2×1)=10/2=5.

f(x)=x^2 -10x +24

f(5)=5^2 -10 (5)+24=25 -50 +24=-1

Hence, the vertex of the parabola will be at (5,-1).

The point (0,c) will be the y-intercept of the parabola. So, (0,24).

Now, let us use the information to plot the graph.

Vertex of the parabola will be at (5,-1). And, the axis of symmetry would be at x=5. And y-intercept of the parabola would be at (0,24).

The graph of the function crosses the x-axis at two points, so it will have two solutions.
Take a break, have some water and take yourself out of laser mode. 

You have paid great attention to this and it is time to rewind all the learnings in your head. If you enjoyed learning and solving the above three equations. Then congratulations you now don’t need to ask anyone “what is the graphing method” ever again. 

Also, you are one step closer to mastering the method of solving quadratic equations by graphing technique. But as everything in life requires practice and focus, so will you! This blog must have helped you in understanding the basics and a little complex equations along with their answers. 

Hence listed a few common quadratic equations that you can try to graph the following quadratic equations on your own:

  • X^2−3x−10= 0 
  • 6x – 9 – x^2 = 0 
  • x^2 + 2x – 15 = 0 
  • x^2 – 8x + 15 = 0 
  • 2x^2 + 6x + 3 = 0

Did you get stuck or need more practice on this method then simply talk to an expert at TutorEye and we can connect you to a math tutor in under 3 minutes. By taking personalized learning sessions on solving quadratic equations, our tutors can help you get the A+ grades that you have been dreaming of achieving. We also offer online tutoring classes in other core subjects like Science, Chemistry, Biology, Physics etc. 

Our tutoring sessions are designed to fit your life and schedule, and hence, we are available 24/7 to help you learn any topic, get ready for an exam, or advance your skill set. 

If you are restricted by budget then also, we have a solution for you; you can refer to our free math tutorial available on YouTube on solving quadratic equations by graphing method and many more topics like these! 

So, why hold to your big dreams? Take some extra help from experts and get started with e-learning asap!

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