Question

Write the Quadratic Formula for the quadratic equation.

Answer 1

`x = (-b \pm √(b^2 - 4ac))/(2a)`

Explanation:

Hello!

Let's start with the given equation and work our way towards x.

Solve for x

• 
  `ax^2 + bx + c = 0`

Divide both sides by a

• 
  `a(x^2 + \frac bax + \frac ca) = 0`
• 
  `x^2 + \frac bax + \frac ca = 0`

Move c/a to the other side

• 
  `x^2 + \frac bax = -\frac ca`

At the step, we have to use the Completing the Square method.

• 
  `x^2 + \frac bax + ((b)/(2a))^2 = -\frac ca + (\frac b{2a})^2`
• 
  `x^2 + \frac bax + (b^2)/(4a^2) = -\frac ca + \frac {b^2}{4a^2}`
• 
  `(x + \frac b{2a})^2= -\frac ca + (b^2)/(4a^2)`

Multiply -c/a by 4a and add the two fractions

• 
  `(x + \frac b{2a})^2= (b^2 - 4ac)/(4a^2)`

Square Root both sides

• 
  `\sqrt{(x + \frac b{2a})^2}= \sqrt{(b^2 - 4ac)/(4a^2)}`
• 
  `x + \frac b{2a} = \pm(√(b^2 - 4ac))/(2a)}`

Subtract b/2a

• 
  `x = \pm ( √(b^2 - 4ac))/(2a) - \frac b{2a}`
• 
  `x = (-b \pm √(b^2 - 4ac))/(2a)`

And that's the derivation of the Quadratic Formula.