Saturday, December 15, 2018

Quadratic Equation

A quadratic equation in the variable x is an equation of the form ax2 + bx + c = 0, where a, b, c are real numbers, a ≠ 0. For example x2 + 4x + 4 = 0, x2 - 9 = 0

Standard form of a quadratic equation.
ax2 + bx + c = 0, where a, b, c are real numbers, a ≠ 0

In general, a real number α is called a root of the quadratic equation ax2 + bx + c = 0, a ≠ 0 if a α2 + bα + c = 0. We also say that x = α is a solution of the quadratic equation, or that α satisfies the quadratic equation. 

The zeroes of the quadratic polynomial ax2 + bx + c and the roots of the quadratic equation ax2 + bx + c = 0 are the same.

Method of completing the square
example 2x2 – 7x + 3 = 0

2x2 – 7x = - 3
On dividing both sides of the equation by 2, we get
x2 – 7x/2 = -3/2
x2 – 2 * x * 7/4 = -3/2
On adding (7/4)2 to both sides of equation, we get
(x)2 - 2 * x * 7/4 + (7/4)2 
= (7/4)2 - 3/2
(x - 7/4)2 = 49/16 - 3/2
(x - 7/4)2 = 25/16
(x - 7/4) = ± 5/4
x = 7/4 ± 5/4
x = 7/4 + 5/4 or 
x = 7/4 - 5/4
x = 12/4 or x = 2/4
x = 3 or 1/2

Method of finding roots by quadratic formula
2x2 – 7x + 3 = 0

On comparing this equation with ax2 + bx + c = 0, we get
a = 2, b = -7 and c = 3
By using quadratic formula, we get
x = -b ± √b2 - 4ac/2a
x = (7±√49 - 24)/4
x = (7±√25)/4
x = (7±5)/4
x = 7+5/4 or 
x = 7-5/4
x = 12/4 or 2/4
∴ x = 3 or 1/2

Nature of roots
We know that the roots of the equation ax2 + bx + c = 0 are given by
quadratic formula,
x = -b ± √b2 - 4ac/2a

If b2 – 4ac > 0, we get two distinct real roots
-b + √b2 - 4ac/2a

and
-b - √b2 - 4ac/2a

If b2 – 4ac = 0, then x = -b/2a
We can say that the quadratic equation ax2 + bx + c = 0 has two equal real roots in this case

If b2 – 4ac < 0, then there is no real number whose square is b2 – 4ac.
∴ There are no real roots for the given quadratic equation in this case.

Since b2 – 4ac determines whether the quadratic equation ax2 + bx + c = 0 has real roots or not, b2 – 4ac is called the discriminant of this quadratic equation.

A quadratic equation ax2 + bx + c = 0 has
(i) two distinct real roots, if b2 – 4ac > 0,
(ii) two equal real roots, if b2 – 4ac = 0,
(iii) no real roots, if b2 – 4ac < 0.

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