## Complex Numbers and Quadratic Equations

# Quadratic Equations

**Tips:**

- Roots of the quadratic equation ax
^{2}+ bx + c = 0 are \tt x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} - If b
^{2}− 4ac > 0 then the roots are real and distinct

If b^{2}− 4ac < 0 then the roots are imaginary

If b^{2}− 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 ax
^{4}+ bx^{3}+ cx^{2}+ 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 ax
^{2}+ 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) = ax
^{2}+ 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 ax
^{2}+ 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 ax
^{2}+ bx + c = 0 may be in the ratio m:n is mnb^{2}= ac(m+n) - α
^{4}+ β^{4}= (α^{3}+ β^{3})( α + β) – αβ (α^{2}+ β^{2}) - α
^{5}+ β^{5}= (α^{3}+ β^{3})(α^{2}+ β^{2}) – α^{2}β^{2}(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 ax
^{2}+ 2hxy + by^{2}+ 2gx + 2fy + c = 0 in ‘x’ and ‘y’ may be resolved into two linear rational factors if abc + 2fgh – af^{2}– bg^{2}– ch^{2}= 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 a
_{0}+ a_{1}x + a_{2}x^{2}+ …….. a_{n}.x^{n}= 0 is zero, then x = 1 is always atleast one root of equation. - Length of latus rectum of parabola y = ax
^{2}+ bx + c is \tt \frac{1}{|a|}

### Part1: View the Topic in this video From 00:40 To 51:48

### Part2: View the Topic in this video From 00:40 To 52:20

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

Disclaimer: Compete.etutor.co may from time to time provide links to third party Internet sites under their respective fair use policy and it may from time to time provide materials from such third parties on this website. These third party sites and any third party materials are provided for viewers convenience and for non-commercial educational purpose only. Compete does not operate or control in any respect any information, products or services available on these third party sites. Compete.etutor.co makes no representations whatsoever concerning the content of these sites and the fact that compete.etutor.co has provided a link to such sites is NOT an endorsement, authorization, sponsorship, or affiliation by compete.etutor.co with respect to such sites, its services, the products displayed, its owners, or its providers.

1. **Quardratic Equation** ax^{2} + 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}

2. **Quadratic Equation**

If roots of quadratic equation ax^{2} + 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 ax^{3} + bx^{2} + 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 ax^{2} + 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) α^{3} + β^{3} = (α + β)^{3} − 3αβ(α + β) = \tt -\frac{b\left(b^2-3ac\right)}{a^3}

e) α^{3} − β^{3} = (α − β)^{3} + 3αβ(α − β) = \tt \frac{\pm\left(b^{2}-ac\right)\sqrt{b^{2}-4ac}}{a^{3}}

f) α^{4} + β^{4} = {(α + β)^{2 }− 2αβ}^{2 }− 2α^{2} β^{2} = \tt \left(\frac{b^{2}-2ac}{a^{2}}\right)^2-\frac{2c^{2}}{a^{2}}

g) α^{4} − β^{4}=(α^{2} − β^{2})(α^{2} + β^{2}) = \tt \frac{\pm b\left(b^{2}-2ac\right)\sqrt{b^{2}-4ac}}{a^{4}}

h) α^{2}+ αβ + β^{2} = (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) α^{2}β + β^{2}α = 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}}