751 views

### Can you derive the Quadratic Formula?

For a general quadratic equation of the form ax2 + bx + c = 0 (1)
We will solve for x via the ‘Completing the Square‘ method:

First we divide equation (1) through by a on both sides, yielding:
x2 +(b/a)x + c/a = 0

Slightly rearranging, we then write (for tidiness):
x2 + (b/a)x = -c/a (2)

Now the key behind completing the square is to try and write the x2 and x terms as the square of some quantity, which is close to:
(x + (b/2a))2 (3)

What we have done is half the coefﬁcient of the x term, as when the brackets are expanded out, you have a certain sum repeated twice, I will demonstrate in fuller detail now what I mean.

Expanding out (3), we have
(x +(b/2a))(x+(b/2a)) = x2+ (b/2a)x + (b/2a)x + (b2/4a2= x2 + (b/a)x + (b2/4a2) , which is almost identical to the left hand side of equation (2), except now we have the extra third term which is (b2/4a2)

We can rearrange this easily to show that x2+ (b/a)x =(x +(b/2a))2-(b2/4a2)
Substituting this identity into equation (2), we now have
(x +(b/2a))2 -(b2/4a2) = -c/a (4)

We now rearrange as follows:
(x+(b/2a))2 = (b2/4a2) - (c/a) = (b2-4ac)/(4a2) ,
where in the third equality I have simply summed over a common denominator by multiplying the second term by 4a over the numerator and denominator.

Finally we take the square root of both sides, and rearrange to arrive at the quadratic formula we were looking for, as follows:
x = (-b ± sqrt(b2-4ac))/2a

Note that we have two solutions to the quadratic equation, resulting from the existence of both positive and negative square roots, hence the ± sign. Also ‘sqrt' denotes taking the square root.

Corollary

Note that the solution of x depends on the quantity sqrt(b2-4ac), which is known as the Discriminant. This is important as it results with 3 distinct cases for the solution which are as follows:

1) b2 - 4ac > 0 , and so it gives one positive and one negative square root, leading to 2 unique solutions.
2) b2 -4ac = 0, which then yields exactly one solution, known as a repeated root, -b/2a .
3) b2 -4ac < 0, which cannot be square rooted within our common number system which we call the Real Numbers. If we extend our number system, to allow for square roots of negative numbers, we call these the Complex Numbers, which go far beyond the scope of the course.

If you want to ﬁnd out more, be sure to stay on till A Level and take Further Maths!

3 years ago

Answered by Kamran, who tutored GCSE Maths with MyTutor

## Still stuck? Get one-to-one help from a personally interviewed subject specialist

#### 628 SUBJECT SPECIALISTS

£20 /hr

Degree: Mathematics (Bachelors) - Imperial College London University

Subjects offered:Maths

Maths

“21 years old. 2 years experience teaching. BSc Mathematics Imperial College London. Studying MSc in Aerospace Dynamics (please ask for references).”

£20 /hr

Degree: Electrical and Electronic Engineering (Masters) - Imperial College London University

Subjects offered:Maths, Physics+ 1 more

Maths
Physics
Chemistry

“I am passionate, motivated, and always hungry for more knowledge. I can bring energy and life to the classroom, and engage the student.”

£18 /hr

Degree: BA Biological Sciences (Bachelors) - Oxford, Magdalen College University

Subjects offered:Maths, Physics+ 3 more

Maths
Physics
German
Chemistry
Biology

“Chatty, patient and enthusiastic Oxford Biology student. I did my A Levels just last year (A*A*A*A), so I can share my revision experience to help you achieve!”

Currently unavailable: for new students

Degree: Mathematics (Masters) - Bristol University

Subjects offered:Maths, Physics+ 1 more

Maths
Physics
Further Mathematics

“Mathematics masters student from the University of Bristol. Passionate about my subject, looking to use it to help boost your grades.”

MyTutor guarantee

### You may also like...

#### Other GCSE Maths questions

A teacher is chosen at random. The probability that the teacher is female is 3/5. There are 36 male teachers. How many teachers in total work at the school?

How do I find the equation of a line connecting points a(p,q) and b(r,s)?

How is trigonometry used on non-right angled triangles?

How do I find the equation of a line that is perpendicular to another line?

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this.
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}