The theory of quadratic equation formulae will help us to solve different types of problems on quadratic equation.
The general form of a quadratic equation is ax\(^{2}\) + bx + c = 0 where a, b, c are real numbers (constants) and a ≠ 0, while b and c may be zero.
(i) The Discriminant of a quadratic equation is ax\(^{2}\) + bx + c = 0 (a ≠ 0) is ∆ = b\(^{2}\) - 4ac
(ii) If α and β be the roots of the equation ax\(^{2}\) + bx + c = 0 (a ≠ 0) then
α + β = -\(\frac{b}{a}\) = -\(\frac{coefficient of x}{coefficient of x^{2}}\)
and αβ = \(\frac{c}{a}\) = \(\frac{constant term}{coefficient of x^{2}}\)
(iii) The formula for the formation of the quadratic equation
whose roots are given: x^2 - (sum of the roots)x + product of the roots = 0.
(iv) When a, b and c are real numbers, a ≠ 0 and discriminant is positive (i.e., b\(^{2}\) - 4ac > 0), then the roots α and β of the quadratic equation ax\(^{2}\) + bx + c = 0 are real and unequal.
(v) When a, b and c are real numbers, a ≠ 0 and discriminant is zero (i.e., b\(^{2}\) - 4ac = 0), then the roots α and β of the quadratic equation ax\(^{2}\) + bx + c = 0 are real and equal.
(vi) When a, b and c are real numbers, a ≠ 0 and discriminant is negative (i.e., b\(^{2}\) - 4ac < 0), then the roots α and β of the quadratic equation ax\(^{2}\) + bx + c = 0 are unequal and imaginary. Here the roots α and β are a pair of the complex conjugates.
(viii) When a, b and c are real numbers, a ≠ 0 and discriminant is positive and perfect square, then the roots α and β of the quadratic equation ax\(^{2}\) + bx + c = 0 are real, rational unequal.
(ix) When a, b and c are real numbers, a ≠ 0 and discriminant is positive but not a perfect square then the roots of the quadratic equation ax\(^{2}\) + bx + c = 0 are real, irrational and unequal.
(x) When a, b and c are real numbers, a ≠ 0 and the discriminant is a perfect square but any one of a or b is irrational then the roots of the quadratic equation ax\(^{2}\) + bx + c = 0 are irrational.
(xi) Let the two quadratic equations are a1x^2 + b1x + c1 = 0 and a2x^2 + b2x + c2 = 0
Condition for one common root: (c1a2 - c2a1)^2 = (b1c2 - b2c1)(a1b2 - a2b1), which is the required condition for one root to be common of two quadratic equations.
Condition for both roots common: a1/a2 = b1/b2 = c1/c2
(xii) In a quadratic equation with real coefficients has a complex root α + iβ then it has also the conjugate complex root α - iβ.
(xiii) In a quadratic equation with rational coefficients has a irrational or surd root α + √β, where α and β are rational and β is not a perfect square, then it has also a conjugate root α - √β.
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