Quadratic Equations (Roots)

We have come across quadratic equations in the algebra section and when were simply plotting quadratic equations on a 2 dimensional graph. It is important before you read this page that you understand the quadratic formula page, completing the square page and the plotting quadratic equations page.
THERE IS ACTUALLY NOTHING HARD ON THIS PAGE IF YOU HAVE UNDERSTOOD THESE 3 PAGES.

What are the ROOTS of an equation?

When we plot quadratic equations, some equations would cross the x axis. What does it mean if the curve crosses the x axis like in figure 1. Well, it means that the point at which the curve crosses the axis would be in the form of (x,0). Remember any point on the x-axis is in the form of (x,0), because the y-value would always be 0.
So in figure 1 we have the equation y = x² + 3x - 4. We can see that when we plot the equation the curve crosses the x-axis at the points (-4,0) and (1,0). Remember that on a 2 dimensional graph, we plot the inputs on the x-axis (horizontal axis) and the outputs on the y-axis (vertical axis). So the points (-4,0) and (1,0) tell us that when x is equal to -4 and 1, the output of the quadratic equation would equal to 0. Lets show this.
When x = -4, y = (-4)² + 3(-4) - 4 = 16 - 12 - 4 = 0When x = 1, y = (1)² + 3(1) - 4 = 1 + 3 - 4 = 0
So -4 and 1 are the ROOTS OF THE EQUATION y = x² + 3x - 4. A root of any equation are the input values which make the output value of an equation equal to 0.
By looking at a graph of a quadratic equations you can identify, how many roots a quadratic equation has (in our case we have 2 roots which are -4 and 1) AND what are the roots of the quadratic equation (in our case the roots of y = x² + 3x - 4 are -4 and 1).

Quadratic Formula

How do we know when a quadratic function has roots?

Lets say we did not have a graph of the quadratic equation, how would we find the roots of the equation? There is an option of plotting the graph to see whether the quadratic equation crosses the x-axis or not, OR we can find the roots using what we have learned in the past.
Before we try and find the roots of a quadratic equation, lets first find whether the quadratic equation has roots or not. Then we can go through on how to find the roots, given we know that the quadratic equation has roots. We can do this by using a part of the quadratic formula as discussed below.
Remember that the quadratic formula is used to find value of x when the equation is equal to 0. So essentially by using the quadratic formula, we can find the values of x which satisfies the equation when equal to 0. If we are able to find values of x using the quadratic formula, it will well tells how many roots and what the roots actually are for the equation, as the values of the input (in this case x) which makes the value of the output (in this case y) equal 0 are the roots.
We still have not answered the question though, how do we know whether a quadratic equation has roots, without working out what the roots are?
Well, you can see in the quadratic formula we take the square root of b²- 4ac.
If the value of b²- 4ac is positive and not equal to 0, then we would find two values of x (the input), thus the quadratic equation will have two roots. This is because the square root of a positive number be a negative and positive number which that is why the quadratic formula has a plus-minus sign.
If the value of b²- 4ac is equal to 0, then we would find one value of x (the input), thus the quadratic equation will have one root. This is because the square root of 0 is 0.
If the value of b²- 4ac is negative, then we would find zero values of x (the input), thus the quadratic equation will have 0 roots. This is because we cannot take the square root of a negative number, thus there is no value of x (the input) which makes the quadratic equation equal 0.
By the way, b²- 4ac is called THE DISCRIMINANT (that is just the name given to the expression). So we can say, to find how many roots a quadratic equation has, we can use the discriminant and thus the discriminant's value.
So when we are given a quadratic equation and we need to find the roots of the quadratic equation, we can first evaluate whether the quadratic equation has roots or not and then work out the values of the roots. How we work out the values of the roots is discussed below.

Finding the roots of quadratic equations

One way is by using the quadratic formula. The quadratic formula would give us the values of x (the input) which makes the output of the quadratic expression equal to 0. Thus, we are able to identify the input values, as explained in the quadratic formula page to find the roots.
Another way is to factorise the quadratic expression. The benefit of factorising is that it gives us another way of writing the quadratic expression. When we need to find the to find the values of x (the input) when the quadratic expression is equal to 0. Thus, we are able to identify the input values, as explained in the factorising (2 Brackets) pages to find the roots.
Another way is to complete the square. Rearrange the quadratic expression by completing the square and then solve to find the value of the input, when the rearranged expression is equal to 0. The input values which we get are the roots.
(A SUGGESTION IS THAT WHEN YOU ARE TRYING TO FIND THE ROOTS OF A QUADRATIC EQUATION, CHECK FIRST WHETHER THE QUADRATIC EQAUTION HAS ROOTS BY USING THE DISCRIMINANT.

Page updated

Google Sites

Report abuse