What is a quadratic equation?
A quadratic equation is an equation that can be rearranged as a standard form as ax2 + bx + c = 0, here a ≠ 0, x is an unknown number also a, b & c are known numbers
If a = 0, then standard equation become bx + c = 0, this is a linear equation with solution x = -c/b not a quadratic equation.
We can also represent a quadratic equations in a factored form then,
ax2 + bx + c = 0 = (x-m)(x-n), here m & n are roots of the quadratic equation
How to solve a Quadratic equation?
A quadratic equation has at least one root and at most two roots.
If b2 – 4ac = 0 or b = 0 then the quadratic equation has only one root, If b2 – 4ac ≠ 0 & b ≠ 0 then quadratic equation has two roots.
If b2 – 4ac > 0 then quadratic equation has only real roots, If b2 – 4ac < 0 then quadratic equation has only complex roots
1) Solving quadratic equation by completing the square
We know a quadratic equations is in the form of ax2 + bx + c = 0. We can transform this equation in to (x + u)2 = v
Let consider, ax2 + bx + c = 0
ax2 + bx = -c. Divided both sides with a then,
x2 + bx/a = -c/a, now add square of the half of the coefficient of x, that is (b/2a)2
x2 + bx/a + (b/2a)2 = -c + (b/2a)2.
We know (r + s)2 = r2 + 2rs + s2
so, (x + b/2a)2 = -c + (b/2a)2. Let u = b/2a and v = -c + (b/2a)2 then
(x + u)2 = v
take square roots
x + u = ±√v
⇒ x = -u ± √v
Example problems using completing the square
Q. solve the equation x2 + 5x + 6 = 0
x2 + 5x = -6
⇒ x2 + 5x + (5/2)2= -6 + (5/2)2
then, (x + 5/2)2= -6 + 25/4 = ¼
x + 5/2 = ±½
so, x = ±½ – 5/2
⇒ x = -2 and x = -3
2) Solving quadratic equation by factorisation
We Know a quadratic equations can be represented as a factorised form that is
ax2 + bx + c = 0 = (x-m)(x-n) = x2 – (m+n)x + mn here – (m+n) = b and c = mn
We guess two numbers (m & n) that satisfy the equation for a and b then these two numbers are the roots of the quadratic equation
Example problems using factorisation
Q. solve the equation x2 + 5x + 6 = 0
From above equation we get b = -(m + n) = 5 and c = 6 = mn
Here we can guess m = -2 and n = -3 so solution of the equation are -2 & -3
3) Solving with quadratic formula
Quadratic formula is a formula which gives solution of the quadratic equations
We can derive x from the quadratic equations that is,
ax2 + bx + c = 0
⇒ ax2 + bx = – c
so, x2 + bx/a = – c/a
⇒ x2 + bx/a + (b/2a)2 = – c/a+ (b/2a)2
thus, (x + b/2a)2 = – 4ac/4a2+ b2/4a2 = (b2 – 4ac)/4a2
x + b/2a = ±√(b2 – 4ac)/2a
x = (-b ±√(b2 – 4ac))/2a
Example problems using factorisation
Q. solve the equation x2 + 5x + 6 = 0
x = (-5 ±√(52 – 4×1×6))/2×1
⇒ x = (-5 ±√(25 – 24))/2
⇒ x = (-5 ± 1)/2
then, x = -2 & x = -3
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