Can every quadratic equation be solved? Does a quadratic equation always have more than one solutions? Are there any equations that don’t have any real solution? The value of the variable for which the equation gets satisfied is called the solution or the root of the equation. The Nature of Roots of a quadratic equation is very interesting. Let us find out how!
Roots of a Quadratic Equation
The number of roots of a polynomial equation is equal to its degree. Hence, a quadratic equation has 2 roots. Let α and β be the roots of the general form of the quadratic equation ax² + bx + c = 0. We can write:
α = (-b-√b²-4ac)/2a
and β = (-b+√b²-4ac)/2a
Hence, the nature of the roots α and β of equation ax² + bx + c = 0 depends on the quantity or expression (b² – 4ac) under the square root sign. We say this because the root of a negative number can’t be any real number. Say x² = -1 is a quadratic equation. There is no real number whose square is negative. Therefore for this equation, there are no real number solutions.
Hence, the expression (b² – 4ac) is called the discriminant of the quadratic equation ax² + bx + c = 0. Its value determines the nature of roots as we shall see. Depending on the values of the discriminant, we shall see some cases about the nature of roots of different quadratic equations.
•Case I: b² – 4ac greater than 0
When a, b, and c are real numbers, a ≠ 0 and the discriminant is positive, then the roots α and β of the quadratic equation ax² +bx+ c = 0 are real and unequal.
•Case II: b²– 4ac = 0
When a, b, and c are real numbers, a ≠ 0 and the discriminant is zero, then the roots α and β of the quadratic equation ax²+ bx + c = 0 are real and equal, and such Equation is a perfect square quadratic equation.
•Case III: b²– 4ac Less than 0
When a, b, and c are real numbers, a ≠ 0 and the discriminant is negative, then the roots α and β of the quadratic equation ax² + bx + c = 0 are unequal and not real. In this case, we say that the roots are imaginary.
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