Prove why the quadratic formula works

The quadratic formula is x = [-b ± √(b2 - 4ac)]/2a (1) ,where a,b and c are the coefficents of a quadratic in the form ax^2+bx+c=0 (2).

To prove that this works we need to start from equation (2)and using algebra get to equation (1).

So our first step will be to take equation (2) and divide it all by a. This leaves us with x^2+(b/a)x+(c/a)=0 (3). We are allowed to do this as we know that is non zero.

Then we subtract (c/a)from both sides of equation (3), which results in x^2+(b/a)x=-(c/a) (4).

In the next step we want to complete the square so we have to add (b/2a)^2 to both sides of equation (4). This the gives us x^2+(b/a)x+(b/2a)^2=-(c/a)+(b/2a)^2 (5).

We then complete the square on the LHS of (5) to give us [x+(b/2a)]^2=-(c/a)+(b/2a)^2  and here the RHS can als be re-written to give us (b^2-4ac)/2a to give us a full equation of [x+(b/2a)]^2=(b^2-4ac)/2a (6).

Now we can take the square root of both sides of (6) to give us x+(b/2a)=(sqrt(b^2-4ac)/2a) (7).

Now all we have to do is minus (b/2a) from both sides, this results in the quatratic formula of  x = [-b ± √(b2 - 4ac)]/2(1) so there we have proven why the quadratic formula works.

Jake R. A Level Maths tutor, GCSE Maths tutor, 13 Plus  Maths tutor

2 years ago

Answered by Jake, an A Level Maths tutor with MyTutor

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


£24 /hr

Tiarnan B.

Degree: MChem (Masters) - York University

Subjects offered:Maths, Geography+ 2 more

-Personal Statements-

“I am a fun, open and qualified tutor. I guarantee by choosing me you will reach where you want to be and beyond, all while enjoying the subject!”

£36 /hr

Joe B.

Degree: Mathematics G100 (Bachelors) - Bath University

Subjects offered:Maths, Further Mathematics + 4 more

Further Mathematics
-Personal Statements-

“About Me Hi, I'm Joe, a first year mathematics student from Luton studying at Bath University. I am an accomplished mathematician and economist, having achieved A* grades in A Level Maths, Further Maths and Economics in June 2016. As ...”

£22 /hr

Chris S.

Degree: Psychology (Bachelors) - Nottingham University

Subjects offered:Maths, Psychology+ 1 more

-Personal Statements-

“*Over 25 completed MyTutor school tutorials* I am a reliable and enthusiastic maths and psychology tutor.”

About the author

£20 /hr

Jake R.

Degree: Mathematics (Bachelors) - Exeter University

Subjects offered:Maths


“Top tutor from the renowned Russell university group, ready to help you improve your grades.”

MyTutor guarantee

You may also like...

Other A Level Maths questions

what is implicit differentiation and how is it achieved?

Prove why the quadratic formula works

How can we solve a two-equation, two-unknown values?

How would you differentiate f(x)=3x(2x-1)^2

View A Level Maths tutors

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this. Dismiss