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

1.Factorisation
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.

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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.

Example-

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,...

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5.
Solve the equation
1 2 x − x = 3 by the quadratic formula. (Give the answer in surd form.) 2
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