Question

How is the quadratic formula derived from the quadratic equation?

Answer 1

I hope this helps you

Answer 2

Full explanation is down here

Explanation:

Hello!

Recall the standard form of a quadratic:
`ax^2 + bx + c`

Solve for x:

`ax^2 + bx + c = 0`

• Subtract c

`ax^2 + bx = -c`

• Divide all sides by a

`x^2 + (b)/(a)x = (-c)/(a)`

• Complete the square (make sure to add to both sides)

`x^2 + (b)/(a)x + (b^2)/(4a^2) = (-c)/(a) + (b^2)/(4a^2)`

• Factor the Perfect Square Trinomial

`(x+(b)/(2a))^2 = (-c)/(a) + (b^2)/(4a^2)`

• Simplify the right side using a LCD (least common denominator)

`(x+(b)/(2a))^2 = (-4ac+b^2)/(4a^2)`

• Take the square root of both sides

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

• Simplify (make sure to add the plus or minus sign!!)

`x+(b)/(2a) = (\pm√(-4ac+b^2))/(2a)`

• Subtract b/2a from both sides

`x = (\pm√(-4ac+b^2))/(2a) - (b)/(2a)`

• Convert to traditional format

`x = (-b\pm√(b^2 - 4ac))/(2a)`

Completing the square:

Recall the standard form of a quadratic:
`ax^2 + bx + c`

To complete a square, we must find the "c" value given
`ax^2 + bx + c` so that it converts into a perfect square trinomial.

To find c:

• Take the b-value
• Divide it by 2
• Square it

Example: Given the expression
`x^2 - 8x`, complete the square

Let's find the missing "c" value:

• Take the b-value: -8
• Divide it by 2: -4
• Square it: 16

Now, complete the square:

• 
  `x^2 - 8x`
• 
  `x^2 - 8x + 16`

To quickly factor this into a perfect square trinomial, simply take the value you get from the second step (Divide by 2) and plug it in.

Your factored expression will be (x - 4)²