Question

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

Answer 1

Answer:

Quadratic formula can be derived from completing the square method .

Step-by-step explanation:

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

On solving this equation by completing the square , we get

$ax^2+bx+c=0\\x^2+\frac{bx}{a}+\frac{c}{a}=0\\x^2+2\left ( \frac{b}{2a} \right )x+\frac{c}{a}=0\\x^2+2\left ( \frac{b}{2a} \right )x+\left ( \frac{b}{2a} \right )^2-\left ( \frac{b}{2a} \right )^2+\frac{c}{a}=0$

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

On taking square root on both sides, we get

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

which is basically a quadratic formula .

Answer 2

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)