Question

Describe how to derive the quadratic formula from a quadratic equation in standard form

Answer 1

The quadratic formula is derived from a quadratic equation in standard form when solving for x by completing the square. The steps involve creating a perfect square trinomial, isolating the trinomial, and taking the square root of both sides. The variable is then isolated to give the solutions to the equation.

Explanation:

Answer 2

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

Explanation:

The standard form of a quadratic equation in x can be written as;

`ax^(2) +bx+c=0`

where a,b, and c are constants.

The first step is to subtract c on both sides of the equation;

`ax^(2) +bx=-c`

The next step is to divide both sides of the equation by the constant a;

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

Next, we complete the square on the left hand side of the equation by determining another constant c as;

`c=((b)/(2a)) ^(2)=(b^(2) )/(4a^(2) )`

We then add this constant on both sides of the equation in order to complete the square on the L.H.S;

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

The expression on the L.H.S of the equation is now a perfect square and can be factorized to yield;

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

We then take square roots on both sides of the equation and simplify the expression on the R.H.S;

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

The final step is to make x the subject of the formula and a little simplification which will yield the quadratic formula;

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

Putting the expression on the R.H.S under a common denominator yields;

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

which is the quadratic formula