How to Find the Roots of a Quadratic Equation

I will illustrate and describe how to find the roots of a quadratic equation using the quadratic formula. I will also include some extra information, such as the discriminant of the quadratic formula and the proof of the quadratic formula. Enjoy!

Let’s start with the definition of a quadratic equation.

A quadratic equation is an equation in the form ax²+bx+c=0 in which a does not equal 0 and x is an unknown variable. Let’s say our quadratic equation is (1)x²+4x+4=0, and we want to find the roots of the equation. First of all, we must know what a root is. A root is the point where the graph of the equation meets the x-axis. We could graph it out. You could substitute different values for x and find the spot(s) where it meets the x-axis. Or, we could use the quadratic function. The quadratic function is generally denoted as the following:

x=(-b±√(b²-4ac))/2a

Visually, it’s like:

So, If we were to find the roots of the equation x²+4x+4, we would do the following.

(-(4)±(√(4)²-4(1)(4)))/2(1)

=(-4±(√16-16))/2

=(-4±0)/2

=-2

If you were to replace the x in the equation with -2 (edit by author), you would get a true equation. This is a very simple case. Usually, you would get a number under the square root sign, indicating one out of two things. If the number under the square root sign is negative, then your equation does not have any roots. If the number under the square root sign is positive, your equation has two roots. Each root can be found by replacing “+” or “-” for the “±” sign. If the number under the square root sign is 0, as in this case, then your equation has one solution.

Discriminant of the Quadratic Formula

The discriminant of a quadratic formula is the polynomial/expression under the square root sign, which is,

You can use this to find the number of solutions to a quadratic equation. If the discriminant is greater than 0, then the quadratic equation has 2 solutions. For example, x²+5x+6, 25-4(1)(6)=1, which is greater than 0. If the discriminant is 0, then the quadratic equation has 1 solution. For example, the example first given, or x²+6x+9, 36-4(1)(9)=0, which is equal to 0. If the discriminant is less than 0, then the quadratic equation has 0 real roots (solutions), as in 2x²+4x+5, 16-4(2)(5)=-24, which is less than 0.

Let’s do another example:2x²+5x+2

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12 Responses to “How to Find the Roots of a Quadratic Equation”
  1. Roy Says...

    On January 22, 2009 at 10:27 pm

    Very exciting article! But, the visual thing is a little messed up. I’d give a 97/100!!!!


  2. Unanimous Says...

    On February 1, 2009 at 6:08 pm

    Perfect!


  3. stevil kinevil Says...

    On February 7, 2009 at 9:04 am

    The solutions can also be found by factorising, in this case the equation can be simplified to (x+2)(x+2)=0 which therefor means x has to be 2. Still a nice article tho. And i should point out that if the number under the square root it zero, it doesnt mean there are no solutions, it means there are no real solutions. There are imaginary solutions, which involve i, which is the root of negative 1.


  4. razorsharp1 Says...

    On February 11, 2009 at 11:52 pm

    Thanks for the tip, I’ll try to learn…


  5. anonymous Says...

    On March 17, 2009 at 11:00 pm

    Hey I still dont understand… How do you know that you have to subsitute x with 4. Is it a trial and error thing or a guess or is there a special method?
    I have a question with me whereby it only provides the roots and I am supposed to find the other a, b and c. How should I go about finding it?


  6. razorsharp1 Says...

    On March 18, 2009 at 6:04 pm

    Sorry there’s an error in the document; the “4″ should be replaced with a “-2″, the answer we got for x via the quadratic formula. -2 would work when substituted in the original equation, x^2 + 4x + 4.

    As for your other question, if you know the root(s), there are an infinite possible ways to arrange a, b, and c to fit with the roots. A property that is true with this: If the roots are 4 and 6, find the average, which is 5 (which is also the axis of symmetry for the parabola). That means that (-b)/(2a)=5. “c” is the y-intercept, as in a linear equation. Think of it this way: there is more than one parabola that intersects with 2 points; in this case, the x-intercept(s). Even if you only have 1 or 0 roots, the same things apply.


  7. cheche Says...

    On September 25, 2009 at 5:18 am

    tnx for the info…
    perfect!!


  8. Filip Says...

    On December 14, 2009 at 7:24 pm

    You saved me! :)


  9. rj Says...

    On February 5, 2010 at 9:24 am

    so what is the roots?


  10. razorsharp1 Says...

    On February 5, 2010 at 9:33 pm

    If you mean x^2 + 4x + 4, then there is only one root, since -4+0=-4 and -4-0=-4. I’ll edit the article to add some more info.


  11. sweety Says...

    On February 22, 2011 at 8:19 am

    how we find no. of roots?


  12. razorsharp1 Says...

    On January 17, 2012 at 7:37 pm

    Since this is a quadratic, it has a degree of 2. (The degree is the highest exponent that appears, in this case, 2) The number of roots is always equal to the degree, although the roots may not always be real (as stated in above comments). The roots also may not have to be distinct, as in (x+2)(x+2). The number of times a root appears is called the “multiplicity” of that root. This entire “# of roots=degree of polynomial” is called the Fundamental Theorem of Algebra, and it has many, many uses in higher math.

    Hope that helps even though this comment is like a year late.


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