Quadratic Equations

Saturday 25 February 2012

Quadratic Equations

$ax^2+bx+c=0$
Linear Equation can have degree of atmost 1 and has only one solution. Quadratic Equation can have degree of atmost 2 and has two solutions.
General form of quadratic equation: ax² + bx + c = 0, where a, b and c are constants. Note that maximum degree of x is 2.

Solving Quadratic Equations

Unlike linear equations, any quadratic equation always has two solutions called roots of quadratic equation. After solving quadratic equation, you will get two values of x. To solve quadratic equation, you can use directly quadratic formula.

$x={-b\pm\sqrt{b^2-4ac} \over 2a}$

Discriminant

In the above quadratic formula, the expression underneath the square root sign is called the discriminant of the quadratic equation. Discriminant is used to find the nature of roots.

$\Delta = b^2 - 4ac$

Case 1: $\Delta > 0$

Real and distinct roots

Case 2: $\Delta = 0$

Real and one distict root (two same roots)

Case 3: $\Delta < 0$

Roots are imaginary and occur as complex conjugates of each other

Sum and Product of Roots

Let $\alpha$ and $\beta$ be the roots of quadratic equation $x^2+px+q=0$
$x^2+px+q=(x-\alpha)(x-\beta)$
$x^2+px+q=x^2-(\alpha+\beta)x+\alpha \beta$
$p=-(\alpha+\beta)$
$q=\alpha \beta$
$\text{Sum of the roots} = -p$

$\text{Product of roots} = q$ NEXT

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Saturday 25 February 2012