Question

Use complete sentences to explain how the quadratic formula is related to the process of completing the square.

Answer 1

Quadratic formula is derived from completing the square:

ax² + bx + c = 0
ax² + bx = −c
x² + b/a x = −c/a

Complete square on left side by adding (b/(2a))² to both sides:

x² + b/a x + (b/(2a))² = (b/(2a))² − c/a
(x + b/(2a))² = (b²−4ac)/(2a)²
x + b/(2a) = ± √(b²−4ac)/(2a)
x = −b/(2a) ± √(b²−4ac)/(2a)
x = (−b ± √(b²−4ac)) / (2a)

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or

ax² + bx + c = 0
ax² + bx = −c
4a (ax² + bx) = −4ac
4a²x² + 4abx = −4ac

Complete the square on left side by adding b² to both sides

4a²x² + 4abx + b² = b²−4ac
(2ax + b)² = b²−4ac
2ax + b = ± √(b²−4ac)
2ax = −b ± √(b²−4ac)
x = (−b ± √(b²−4ac)) / (2a)

Answer 2

Quadratic formula can be derived from completing the square method .

Explanation:

Consider a quadratic equation :
`ax^2+bx+c=0` where a , b , c are variables and
`a\\eq 0`

On solving this equation by completing the square , we get

`ax^2+bx+c=0\\x^2+(bx)/(a)+(c)/(a)=0\\x^2+2\left ( (b)/(2a) \right )x+(c)/(a)=0\\x^2+2\left ( (b)/(2a) \right )x+\left ( (b)/(2a) \right )^2-\left ( (b)/(2a) \right )^2+(c)/(a)=0`

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

On taking square root on both sides, we get

`\left ( x+(b)/(2a) \right )=\pm \sqrt{(b^2-4ac)/(4a^2)}\\\left ( x+(b)/(2a) \right )=\pm \sqrt{(D)/(4a^2)}\,\,;\,\,D=b^2-4ac\\\left ( x+(b)/(2a) \right )=\pm (√(D))/(2a)\\x=(-b\pm √(D))/(2a)`

which is basically a quadratic formula .