Today is our topic of discussion Quadratic equations of one variable and their solutions .
Quadratic equations of one variable and their solutions
We know, values of variables for which both sides of an equation are equal, are called roots of that equation and these values satisfy the equation.
Linear and quadratic equations of one variable and linear equations of two variables have been discussed in details in the Secondary Algebra Book. A quadratic equation of one variable e.g, ax²+ bx + c = 0 can be solved easily after factorizing the left hand side of the equation if the roots are rational. But all expressions cannot be factorized easily. That is why the following procedure is used to solve the quadratic equations of any form.
Now we solve the 2nd quadratic equation.
ax²+ bx + c = 0
or, a²*x²+ abx + ac = 0 [Multiplying both sides by a]
In equation (1) above, b² – 4ac is called the Discriminant of the quadratic equation, because it determines the state and nature of the roots of the equation.
Variations and nature of the roots of a quadratic equation depending on the conditions of the discriminant
Let a, b, c are rational number. Then,
1) If b²- 4ac > 0 and is a perfect square, then the two roots of the equation are real, unequal and rational.
2) If b²- 4ac > 0 but is not a perfect square, then the two roots of the equation are real, unequal and irrational.
3) If b²- 4ac = 0 hence the roots of the equation are real and equal. In this case, x = – b/2a
4) If b²- 4ac < 0 then the roots of the equation are not real. In this case, the two roots are always conjugate complex or imaginary to each other.
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