Question

1. Derive the quadratic formula from ax^2 + bx + C = 0 by completing the square.

Answer 1

Given:

$ax^2+bx+c=0$

Required:

To derive the quadratic formula.

Explanation:

Consider the given equation,

$ax^2+bx+c=0$

Divide all terms by a, we get

$x^2+\frac{b}{a}x+\frac{c}{a}=0$

Subtract the constant term from both sides of the equation,

$x^2+\frac{b}{a}x=-\frac{c}{a}$

To have a square on the left side the third term (constant) should be

$(\frac{b}{2a})^2$

So add that amount to both sides

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

Re-write the left-side as a square.

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

Take the square root of both sides, we get

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

Now,

$x=\pm\sqrt{(\frac{b}{2a})^2-\frac{c}{a}}-\frac{b}{2a}$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Final Answer:

The quadratic formula is

$x=\frac{-b\pm b^{2}-4ac}{2a}$