Derivation of the formular for Quadratic Equation

What is Quadratic Equation?

Quadratic equation refers to polynomial equation that have a general form of ax²+bx+c=0, where a, b and c are co-efficient. a≠0 otherwise it would be a linear equation and c is constant. The quadratic formula is defined as x= (-b±√b²-4ac)/2a, where “a” is the co-efficient of the x², b is the linear co-efficient of the x and c is the constant term. Therefore, the quadratic formula involves substituting the co-efficient from a given quadratic equation into the formula.

Derivation of the Quadratic Formula

For all quadratic equations, we have the general form:

ax² + bx + c = 0

1. Moving the “non x” to the right,

we get: ax² + bx = – c

2. Dividing by ‘a’ (the coefficient of x²),

we get: x² + bx/a = – c/a

3. We take the coefficient of x, divide it by 2, square the result and then add that to both sides of the equation. The coefficient of x is b/a, one half of that is (b/2a) and squaring that,

we get b²/4a².

Adding that both sides of the equation,

we have x² + bx/a + b²/4a² = – c/a + b²/4a²

4. Taking the square roots of both sides,

we get: x + b/2a = -c/a + b²/4a²

Moving b/2a to the right ,

x =-b/2a±-4ac/4a² + b²/4a²

Simplifying x =-b/2a±b²-4ac/2a

Finally, x=-b±b²-4ac/2a

Assessment

Solve for the roots for the general form of quadratic equation

1. 4x²– 5x + 1 = 0
2. x = 1, ¼ B.  x = -1, ¼  C. 1, 1  D. 1/4, 1
3. 2x²– 14x – 13 = 0
4. 5+53/2, 7+53/2    B. 7+53/2, 5+53/2   C. 7+53/2, 7-53/2 D. 7+56/2, 7+3/2
5. 3x²+ x – 2 = 0
6. x = 2/3, -1 B. x = -2/3, 1  C.  x = -2/3, -1 D. x = -2/3, 1
7. 3×2+4x+1=0
8. x =-1, -1/3 B. x = 1, -1/3 C. x=-1, 1/3 D. x=1, 1/3
9. 2x²– 3x – 6
10. ¾, 4/9   B. ¾, -9/4   C. 9/4, ¾   D. 9/5, ¾