## Complex Numbers and Quadratic Equations

Tips:

• Roots of the quadratic equation ax2 + bx + c = 0 are \tt x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}
• If b2 − 4ac > 0 then the roots are real and distinct
If b2 − 4ac < 0 then the roots are imaginary
If b2 − 4ac = 0 then the roots are real and equal
• A quadratic equation with all odd integer coefficients cannot have rational roots.
• If α, β, γ and δ are roots of the biquadratic equation ax4 + bx3 + cx2 + dx + e = 0, then \alpha + \beta + \gamma + \delta = -\frac{b}{a}\alpha \beta + \alpha \gamma + \alpha \delta + \beta \gamma + \beta \delta + \gamma \delta = \frac{c}{a}, \alpha \beta \gamma + \alpha \beta \delta + \alpha \gamma \delta + \beta \gamma \delta = \frac{-d}{a} and \alpha \beta \gamma \delta = \frac{c}{a}
• If a > 0, then the greatest value will not be there for a quadratic equation ax2 + bx + c = 0, it has a least value \tt \frac{4ac - b^{2}}{4a} \ at \ x = \frac{-b}{2a}. In case of a < 0 then it has a greatest value \tt \frac{4ac - b^{2}}{4a} \ at \ x = \frac{-b}{2a}

Tricks:

• To find the common root of ‘2’ equations, make the coefficient of second degree terms in ‘2’ equations equal and subtract. The value of ‘x’ so obtained is the required common root
• If two quadratic equations with real coefficients have an imaginary root common, then both roots will be common and the ‘2’ equations will be identical. The required condition is \tt \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}
• If ‘2’ quadratic equations have an irrational root common, then both roots will be common and the ‘2’ equations will be identical. The required condition is \tt \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}
• If ‘α’ is a repeated root of the quadratic equation f(x) = ax2 + bx + c = 0 then ‘α’ is also a root of the equation f’(x) = 0
• If ‘α’ is repeated common root of ‘2’ quadratic equations f(x) = 0 and φ(x) = 0, then ‘α’ is also a common root of the equations f’(x) = 0 and φ'(x) = 0.
• In quadratic equation ax2 + bx + c = 0 if a = c both roots are reciprocal to each other
• If a + b + c = 0 then the roots are \tt 1, \frac{c}{a} and if a – b + c = 0 then the roots are −1 and \tt \frac{-c}{a}
• The condition that the roots of the equation ax2 + bx + c = 0 may be in the ratio m:n is mnb2 = ac(m+n)
• α4 + β4 = (α3 + β3)( α + β) – αβ (α2 + β2)
• α5 + β5 = (α3 + β3)(α2 + β2) – α2β2 (α + β)
• \tt |\alpha - \beta| = \sqrt{(\alpha + \beta)^{2} - 4\alpha\beta}
• Let f(x) = 0 be a polynomial equation. Let ‘p’ and ‘q’ be ‘2’ real numbers p < q.
(a) If f(p).f(q)<0, then the equation f(x) = 0 has odd number of real root between p and q.
(b) If f(p).f(q) >0, then the equation f(x) = 0 has either no real root or even number of real roots between ‘p’ and ‘q’
(c) If f(p) = f(q) then the equation f’(x) = 0 has atleast one real root between ‘p’ and ‘q’
• (a) If the coefficients of the polynomial equation f(x) = 0 have 'q’ charges of signs, then the equation f(x) = 0 will have atmost q, positive roots.
(b) If the coefficients of the polynomial equation f(−x) = 0 have p changes of signs, then the equation f(x) = 0 will have atmost of negative roots.
(c) The polynomial equation f(x) = 0 will have atmost p + q real roots where p and q are the changes of signs of coefficients in f(x) and f(−x) [Descarte’s rule of signs]
• The general quadratic expression ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 in ‘x’ and ‘y’ may be resolved into two linear rational factors if abc + 2fgh – af2 – bg2 – ch2 = 0 or \tt \begin{bmatrix}a & h & g \\h & b & f \\g & f & c \end{bmatrix} = 0
• If sum of coefficients of a polynomial equation a0 + a1x + a2x2 + …….. an.xn = 0 is zero, then x = 1 is always atleast one root of equation.
• Length of latus rectum of parabola y = ax2 + bx + c is \tt \frac{1}{|a|}

### Part3: View the Topic in this video From 00:40 To 50:07

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1. Quardratic Equation ax2 + bx + c = 0 (a ≠ 0) has two roots, given by
\tt \alpha = \frac{-b+\sqrt{b^{2}-4ac}}{2a}, \beta = \frac{-b-\sqrt{b^{2}-4ac}}{2a}
or \tt \alpha = \frac{-b+\sqrt{D}}{2a}, \beta = \frac{-b-\sqrt{D}}{2a}

If roots of quadratic equation ax2 + bx + c = 0 (a ≠ 0) are α and β, then
Sum of roots = S = α + β = \tt \frac{-b}{a}=-\frac{coefficient \ of \ x}{coefficient \ of \ x^{2}}
Product of roots = P = α · β = \tt \frac{c}{a}=-\frac{constant \ term}{coefficient \ of \ x^{2}}

3. Cubic Equation
If α, β and γ are the roots of cubic equation ax3 + bx2 + cx + d = 0.
Then, Σ α = α + β + γ = \tt -\frac{b}{a}
Σ αβ = αβ + βγ + γα = \tt \frac{c}{a}
\tt =\alpha \beta \gamma=-\frac{d}{a}

4. Symmetric Roots
If roots of quadratic equation ax2 + bx + c = 0 (a ≠ 0) are α and β, then
a) (α − β) = \tt =\sqrt{\left(\alpha + \beta\right)^{2}-4\alpha \beta}=\pm\frac{b^{2}-4ac}{a}=\frac{+\sqrt{D}}{a}
b) \tt \alpha^{2}+\beta^{2}=\left(\alpha + \beta\right)^{2} -2\alpha \beta =\frac{b^{2}-2ac}{a^{2}}
c) \tt \alpha^{2}-\beta^{2}=\left(\alpha + \beta\right)\sqrt{\left(\alpha+\beta\right)^{2}-4\alpha \beta }=\pm \frac{b\sqrt{b^{2}-4ac}}{a^{2}}=\frac{\pm b\sqrt{D}}{a^{2}}
d) α3 + β3 = (α + β)3 − 3αβ(α + β) = \tt -\frac{b\left(b^2-3ac\right)}{a^3}
e) α3 − β3 = (α − β)3 + 3αβ(α − β) = \tt \frac{\pm\left(b^{2}-ac\right)\sqrt{b^{2}-4ac}}{a^{3}}
f) α4 + β4 = {(α + β)2 − 2αβ}2 − 2α2 β2 = \tt \left(\frac{b^{2}-2ac}{a^{2}}\right)^2-\frac{2c^{2}}{a^{2}}
g) α4 − β4=(α2 − β2)(α2 + β2) = \tt \frac{\pm b\left(b^{2}-2ac\right)\sqrt{b^{2}-4ac}}{a^{4}}
h) α2+ αβ + β2 = (α + β)2 − αβ = \tt \frac{b^{2}-ac}{a^{2}}
i) \tt \frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^{2}+\beta^{2}}{\alpha \beta}=\frac{\left(\alpha +\beta\right)^{2}-2\alpha \beta}{\alpha \beta}=\frac{b^{2}-2ac}{ac}
j) α2β + β2α = αβ(α + β) =  \tt -\frac{bc}{a^{2}}
k) \tt \left(\frac{\alpha}{\beta}\right)^{2}+\left(\frac{\beta}{\alpha}\right)^{2}=\frac{\alpha^{4}+\beta^{4}}{\alpha^{2}\beta^{2}}=\frac{\left(\alpha^{2}+\beta^{2}\right)^{2}-2\alpha^{2}\beta^{2}}{\alpha^{2}\beta^{2}}=\frac{b^{2}D+2a^{2}c^{2}}{a^{2}c^{2}}