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ax² + bx + c = 0

x = (-b ± √(b² - 4ac))/2a


First, rewrite the first equation so that the first coefficient is 1. Divide everything by a.

(ax² + bx + c = 0)/a =

x² + (b/a)x + (c/a) = 0

Isolate (c/a) by subtracting (c/a) from both sides

x² + (b/a)x + (c/a) (-(c/a) = 0 (- (c/a)

x² + (b/a)x = 0 - (c/a)

Add spaces

x² + (b/a)x          =  -c/a

Take 1/2 of the middle term's coefficient and square it. Remember that what you add to one side, you add to the other.

x² + (b/a)x + (b/2a)² = -c/a + (b/2a)²

Simplify the left side of the equation.

x² + (b/a)x + (b/2a)² = (x + (b/2a))²

(x + b/2a))² = ((b²/4a²) - (4ac/4a²)) -> ((b² - 4ac)/(4a²))

Take the square root of both sides of the equation

√(x + b/2a))² = √((b²/4a²) - (4ac/4a²))

x + b/(2a) = (±√(b² - 4ac)/2a

Simplify. Isolate the x.

x = -(b/2a) ± (∛b² - 4ac)/2a = (-b ± √(b² - 4ac))/2a

~

chetankachana

Answer:

Full explanation down here ↓↓↓↓

Step-by-step explanation:

Hello!

Let's start our derivation:

  • [tex]0 = ax^2 + bx + c[/tex]
  • [tex]-c = ax^2 + bx[/tex]

Now, let's complete the square. Set the x² coefficeint to 1 by factoring out a:

  • [tex]-c = a(x^2 + \frac{b}{a}x)[/tex]

Complete the square:

  • Take the "B" value: [tex]\frac ba[/tex]
  • Divide it by 2: [tex]\frac b{2a}[/tex]
  • Square it: [tex]\frac{b^2}{4a^2}[/tex]

Add it to both sides and balance the equation:

  • [tex]-c + \frac{ab^2}{4a^2} = a(x^2 + \frac ba x + \frac{b^2}{4a^2})[/tex]         Balance the equation
  • [tex]-c + \frac{b^2}{4a} = a(x + \frac b{2a})^2[/tex]                    Simplify
  • [tex]-4ac + b^2 = 4a^2(x+\frac b{2a})^2\\[/tex]              Multiply by 4a
  • [tex]\frac{b^2 - 4ac}{4a^2} = (x+\frac b{2a})^2[/tex]                         Divide by 4a²
  • [tex]\sqrt{ \frac{b^2 - 4ac}{4a^2} }= \sqrt{(x+\frac b{2a})^2}[/tex]                 Take the square root of both sides
  • [tex]\frac{\pm\sqrt{b^2 - 4ac}}{2a} = x + \frac b {2a}[/tex]                         Simplify(add plus or minus)  
  • [tex]\frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = x[/tex]                             Simplify

And there you have it! The full proof of the quadratic formula by completing the square.

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