Quadratic equations Any equation of the form ax2 + bx + c=0, where a,b,c are real numbers, a 0 is a quadratic equation.
For example, 2x2 -3x+1=0 is quadratic equation in variable x.
SOLVING A QUADRATIC EQUATION
A real number a is said to be a root of the quadratic equation ax2 + bx + c=0, if aa2+ba+c=0. If we can factorise ax2 + bx + c=0, a 0, into a product of linear factors, then the roots of the quadratic equation ax2 + bx + c=0 can be found by equating each factor to zero.
Example – Find the roots of the equation 2x2 -5x +3=0, by factorisation. Solution:
2x2 -5x +3=0 2x2 -2x-3x+3=0 2x(x-1)-3(x-1)=0 i.e., (2x-3)(x-1)=0 Either 2x-3=0 or x-1=0. So,the roots of the given equation are x=3/2 and x=1.
2. Completing the square
To complete the square means to convert a quadratic to its standard form. We want to convert ax2+bx+c = 0 to a statement of the form a(x h)2 + k = 0.
To do this, we would perform the following steps:
1) Group together the ax2 and bx terms in parentheses and factor out the coefficient a.
2) In the parentheses, add and subtract (b/2a)2, which is half of the x coefficient, squared.
3) Remove the term - (b/2a)2 from the parentheses. Don't forget to multiply the term by a, when removing from parentheses.
4) Factor the trinomial in parentheses to its perfect square form, (x + b/2a)2.
5) Transpose (or shift) all other terms to the other side the equation and divide each side by the constant a.
6) Take the square root of each side of the equation.
7) Transpose the term -b/2a to the other side of the equation, isolating x.
The Quadratic Formula The method of completing the square can often involve some very complicated calculations involving fractions. To make calculations simpler, a general formula for solving quadratic equations, known as the quadratic formula, was derived. To solve quadratic equations of the form ax2 + bx + c = 0,...