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Quadratic formula by completing the square - easier method

By Murray Bourne, 14 Feb 2011

Most math text books derive the Quadratic Formula as follows:

To find the roots of a quadratic equation in the form ax2 + bx + c = 0, follow these steps:

(i) If a does not equal 1, divide each side by a (so that the coefficient of the x2 is 1).

x^2+\frac{b}{a}x+\frac{c}{a}=0

(ii) Rewrite the equation with the constant term on the right side.

x^2+\frac{b}{a}x=-\frac{c}{a}

(iii) Complete the square by adding the square of one-half of the coefficient of x (this is the square of b/2a) to both sides.

x^2+\frac{b}{a}x+\frac{b^2}{4a^2}=\frac{b^2}{4a^2}-\frac{c}{a}

(iv) Write the left side as a perfect square and simplify the right side.

\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}

(v) Equate and solve.

x+\frac{b}{2a}=\pm\sqrt{\frac{b^2-4ac}{4a^2}}

x=-\frac{b}{2a}\pm\frac{\sqrt{b^2-4ac}}{2a}

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

Now, that's pretty messy since there is a lot going on.

Alternative Derivation of Quadratic Formula

Here's a simpler process.
Once again, we start with an equation in the form (and call it Equation [1]):

ax^2+bx+c=0\ \ [1]

Multiply both sides by 4a:

4a(ax^2+bx+c)=4a(0)

4a^2x^2+4abx+4ac=0

Now, go back to the starting equation [1], find the coefficient of x (it's b) and square it (we get b2). Add that number to both sides of our equation:

4a^2x^2+4abx+b^2=b^2-4ac

Write the left side as a perfect square:

(2ax+b)^2=b^2-4ac

Solve for x:

2ax+b=\pm\sqrt{b^2-4ac}

2ax=-b\pm\sqrt{b^2-4ac}

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

I hope you find that easier to follow than the more common method (presented at top).

Solving a quadratic equation using the alternative method of completing the square

Question: Solve the quadratic equation using completing the square:

3x^2-2x-4=0

Answer: In this example. a = 3, so 4a = 12. We multiply both sides by 12:

12(3x^2-2x-4)=0

36x^2-24x-48=0

Add 48 to both sides:

36x^2-24x=48

Now, in the question, b = −2. We square this (b2 = 4) and add it to both sides:

36x^2-24x+4=48+4

Next, write the left side as a perfect square:

(6x-2)^2=52

Solve for x:

6x-2=\pm2\sqrt{13}

6x=2\pm2\sqrt{13}

x=\frac{2\pm2\sqrt{13}}{6}

x=\frac{1\pm\sqrt{13}}{3}

What are your thoughts on this method? Is it easier for you?

[Hat tip to reader Lemmie, who sent me this method.]

See the 14 Comments below.

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