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QUADRATIC EQUATIONS (Part-19) | S.N. Dey Math Solution Series

 

QUADRATIC EQUATIONS (Part-19) | S.N. Dey Math Solution Series


Express each of the following quadratic equations as the difference of two squares and solve :

1. $\,\,x^2+1=0 \\ \Rightarrow x^2-i^2=0 \\ \Rightarrow (x+i)(x-i)=0 \\ \Rightarrow x=-i, i $

2. $\,\, 9x^2+16=0 \\ \Rightarrow (3x)^2-(4i)^2=0 \\ \Rightarrow (3x+4i)(3x-4i)=0 \\ \Rightarrow x=-\frac{4i}{3},\,\,\frac{4i}{3}$

3. $\,\,x^2+x+1=0 \\ \Rightarrow x^2+2.x.\frac 12+(1/2)^2 -(i\frac{\sqrt3}{2})^2=0 \\ \Rightarrow (x+\frac 12)^2- (i\frac{\sqrt3}{2})^2=0  \\  \Rightarrow (x+\frac 12+i\frac{\sqrt3}{2})(x+\frac 12-i\frac{\sqrt3}{2})=0 \\ \Rightarrow x=-\frac 12(1 \pm i\sqrt3)$

$4.\,\,2x^2+2x+5=0 \\ \Rightarrow x^2+x+\frac 52=0 \\ \Rightarrow x^2+2.x.\frac 12+(\frac12)^2+(\frac 32)^2=0 \\ \Rightarrow (x+\frac 12)^2-(i\frac 32)^2=0 \\ \Rightarrow (x+\frac 12+\frac 32 i)(x+\frac 12-\frac 32 i)=0 \\ \Rightarrow x=-\frac 12 \pm i \frac 32$

$5.\,\,3x^2-2x+2=0 \\ \Rightarrow x^2-\frac 23 x+\frac 23=0 \\ \Rightarrow x^2-2.x.\frac 13+(\frac 13)^2 +\frac 59=0 \\ \Rightarrow (x-\frac 13)^2 -(i \frac{\sqrt5}{3})^2=0 \\ \Rightarrow (x-\frac 13+i \frac{\sqrt5}{3})(x-\frac 13-i \frac{\sqrt5}{3})=0 \\ \Rightarrow x= \frac 13 \pm i \frac{\sqrt5}{3}$ 

$6.\,\,8x^2+4x+13=0 \\ \Rightarrow x^2+\frac 12x+\frac{13}{8}=0 \\ \Rightarrow x^2+2.x.\frac 14+(\frac 14)^2 +\frac{25}{16}=0 \\ \Rightarrow (x+\frac 14)^2 -(i \frac 54)^2=0 \\ \Rightarrow (x+\frac 14+\frac 54 i)(x+\frac 14-\frac 54 i)=0 \\ \Rightarrow x=-\frac 14 \pm \frac 54i$

$7.\,\,9x^2+12x+10=0 \\ \Rightarrow x^2+\frac{4}{3}x+\frac {10}{9}=0 \\ \Rightarrow x^2+2.x.\frac 23+(\frac 23)^2 +\frac 23=0 \\ \Rightarrow (x+\frac 23)^2-(i \sqrt{\frac 23})^2=0 \\ \Rightarrow (x+\frac 23)^2-(i \frac{\sqrt6}{3})^2=0 \\ \Rightarrow (x+\frac 23+i \frac{\sqrt6}{3})(x+\frac 23-i \frac{\sqrt6}{3})=0 \\ \Rightarrow x=-\frac 23 \pm i\frac{\sqrt6}{3}$

$8.\,\,5x^2-6x+5=0 \\ \Rightarrow x^2-\frac 65x+1=0 \\ \Rightarrow x^2-2.x.\frac 35+(\frac 35)^2+\frac{16}{25}=0 \\ \Rightarrow (x-\frac 35)^2-(\frac 45 i)^2=0 \\ \Rightarrow (x-\frac 35+\frac 45i)(x-\frac 35-\frac 45i)=0 \\ \Rightarrow x=\frac 35 \pm \frac 45 i$

$9.\,\,a^2x^2-2ax+10=0 \,\,(a \neq 0) \\ \Rightarrow x^2-\frac 2a x+\frac{10}{a^2}=0 \\ \Rightarrow x^2-2.x. \frac 1a+(\frac 1a)^2-(\frac 3a i)^2=0 \\ \Rightarrow (x-\frac 1a)^2-(\frac 3a i)^2=0 \\ \Rightarrow (x-\frac 1a+\frac 3a i)(x-\frac 1a-\frac 3a i)=0 \\ \Rightarrow x=\frac 1a \pm \frac 3a i$

$10.\,\,4x^2-12xp+25p^2=0 \\ \Rightarrow x^2-3xp+\frac{25}{4}p^2=0 \\ \Rightarrow x^2-2.x. \frac 32p+(\frac 32p)^2 -( 2pi)^2=0 \\ \Rightarrow (x-\frac 32 p)^2-(2pi)^2=0 \\ \Rightarrow (x-\frac 32p+2pi)(x-\frac 32p-2pi)=0 \\ \Rightarrow x=\frac 32p \pm 2pi=\frac p2(2 \pm4i)$

11.  State the fundamental  theorem of Algebra :

Any polynomial of degree $\,n\,$ has $\,n\,$ roots.

To get full PDF of S.N. Dey  Math Solutions on Quadratic Equations, click here. 

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