Quadratic Equationsthe
- polynomial of degree 2 is called a quadratic polynomial. Its general form is ax2 + bx + c, where a, b and c are real numbers such that a ≠ 0 and x is a variable quantity.
- If p(x) = ax2 + bx + c, a ≠ 0 is a quadratic polynomial and α is a real number, then p(α) = aα2 + bα + c is known as the value of p(x) at x=α.
- real number α is said to be a zero or root of the quadratic polynomial p(x) = ax2 + bx + c,if p(α) = 0 i.e., aα2 + bα + c = 0 where a ≠ 0 and a, b and c are real numbers.
- If p(x)=ax2 + bx + c is a quadratic polynomial, then p(x)=0, i.e., ax2 + bx + c = 0, a ≠ 0 is called a quadratic equation
- The nature of the roots of the quadratic equation ax2 + bx + c = 0 depends upon the value of D = b2 – 4ac, where D is the discriminant of the quadratic equation
- If ax2 + bx + c, a ≠ 0 is factorizable into a product of two linear factors, then the roots ofthe quadratic equation ax2 + bx + c = 0 can be found by equating each factor to zero.
- The roots of a quadratic equation ax2 + bx + c = 0, where a ≠ 0, can be found by using the formula
x = -b±(√ b2-4ac)/2a provided that b2 – 4ac ≥ 0. - The quadratic equation ax2 + bx + c = 0, a ≠ 0 has :
(i) two distinct real roots, if D = b2 – 4ac > 0
(ii) two equal real roots, if D = b2 – 4ac = 0
(iii) imaginary or no real roots, if D = b2 – 4ac < 0
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