Class 8, Exercise 26, Quadratic Equations

Class 8: Quadratic Equations – Exercise 26

Date: November 28, 2016Author: ICSE CBSE ISC Board Mathematics Portal for Students 0 Comments

Solve each of the following quadratic equations:

Question 1: $x^2-11x+28 = 0$

Answer:

$x^2-11x+28 = 0$

$\Rightarrow$ $x^2 - 7x - 4x + 28 = 0$

$\Rightarrow$ $x (x-4)-7 (x-4) = 0$

$\Rightarrow$ $(x-4)(x-7) = 0$

$\therefore \ x = 4 \ or\ x = 7$

$\\$

Question 2: $x^2-13x = 68$

Answer:

$x^2-13x = 68$

$\Rightarrow$ $x^2-13x-68x = 0$

$\Rightarrow$ $x^2-17x+4x-68x = 0$

$\Rightarrow$ $x(x-17)+4(x-17) = 0$

$\therefore \ x = 17, \ or\ x = -4$

$\\$

Question 3: $x^2+2x = 48$

Answer:

$x^2+2x = 48$

$\Rightarrow$ $x^2+2x-48 = 0$

$\Rightarrow$ $x^2+8x-6x-48 = 0$

$\Rightarrow$ $x(x+8)-6(x+8) = 0$

$\Rightarrow$ $(x-6)(x+8) = 0$

$\therefore \ x = 6 \ or\ x = -8$

$\\$

Question 4: $x^2-2x = 3$

Answer:

$x^2-2x = 3$

$\Rightarrow$ $x^2-2x-3 = 0$

$\Rightarrow$ $x^2-3x+1x-3 = 0$

$\Rightarrow$ $(x+1)(x-3) = 0$

$\Rightarrow$ $x(x-3)+1(x-3) = 0$

$\Rightarrow$ $x = -1 \ or\ x = 3$

$\\$

Question 5: $x^2-49 = 0$

Answer:

$x^2-49 = 0$

$\Rightarrow$ $(x-7)(x+7) = 0$

$\Rightarrow$ $x = 7 \ or\ x = -7$

$\\$

Question 6: $3x^2-243 = 0$

Answer:

$3x^2-243 = 0$

$\Rightarrow$ $3 (x^2-81) = 0$

$\Rightarrow$ $(x-9)(x+9) = 0$

$\Rightarrow$ $x = 9 \ or\ x = -9$

$\\$

Question 7: $16x^2 = 9$

Answer:

$16x^2 = 9$

$\Rightarrow$ $16 x^2-9 = 0$

$\Rightarrow$ $(4x-3)(4x+3) = 0$

$\Rightarrow$ $x =$ $\frac{3}{4}$ $\ or\ x =$ $\frac{(-3)}{4}$

$\\$

Question 8: $3x^2+13x = 10$

Answer:

$3x^2+13x = 10$

$\Rightarrow$ $3x^2+13x-10 = 0$

$\Rightarrow$ $3x^2-2x+15x-10 = 0$

$\Rightarrow$ $x(3x-2)+5(3x-2) = 0$

$\Rightarrow$ $(3x-2)(x+5) = 0$

$\Rightarrow$ $x =$ $\frac{2}{3}$ $\ or\ x = -5$

$\\$

Question 9: $2x^2-7x+3 = 0$

Answer:

$2x^2-7x+3 = 0$

$\Rightarrow$ $2 x^2-6x-x+3 = 0$

$\Rightarrow$ $2x(x-3)-1(x-3) = 0$

$\Rightarrow$ $(2x-1)(x-3) = 0$

$\Rightarrow$ $x =$ $\frac{1}{2}$ $\ or\ x = 3$

$\\$

Question 10: $6x^2+5x = 6$

Answer:

$6x^2+5x = 6$

$\Rightarrow$ $6x^2+5x-6 = 0$

$\Rightarrow$ $6x^2+9x-4x-6 = 0$

$\Rightarrow$ $3x(2x+3)-2(2x+3) = 0$

$\Rightarrow$ $(3x-2)(2x+3) = 0$

$\Rightarrow$ $x =$ $\frac{2}{3}$ $\ or\ x =$ $\frac{(-3)}{2}$

$\\$

Question 11: $10x^2+11x+3 = 0$

Answer:

$10x^2+11x+3 = 0$

$\Rightarrow$ $10x^2+6x+5x+3 = 0$

$\Rightarrow$ $5x(2x+1)+3(2x+1) = 0$

$\Rightarrow$ $(2x+1)(5x+3) = 0$

$\Rightarrow$ $x =$ $\frac{(-1)}{2}$ $\ or\ x =$ $\frac{(-3)}{5}$

$\\$

Question 12: $9x^2+8 = 22x$

Answer:

$9x^2+8 = 22x$

$\Rightarrow$ $9x^2-22x+8 = 0$

$\Rightarrow$ $9x^2-18x-4x+8 = 0$

$\Rightarrow$ $9x(x-2)-4(x-2) = 0$

$\Rightarrow$ $(x-2)(9x-4) = 0$

$\Rightarrow$ $x = 2 \ or\ x =$ $\frac{4}{9}$

$\\$

Question 13: $22x^2+x = 6$

Answer:

$22x^2+x = 6$

$\Rightarrow$ $22x^2+x-6 = 0$

$\Rightarrow$ $22x^2+11x-12x-6 = 0$

$\Rightarrow$ $11x(2x+1)-6(2x+1) = 0$

$\Rightarrow$ $(2x+1)(11x-6) = 0$

$\Rightarrow$ $x =$ $\frac{(-1)}{2}$ $\ or\ x =$ $\frac{6}{11}$

$\\$

Question 14: $2x^2+5x-7 = 0$

Answer:

$2x^2+5x-7 = 0$

$\Rightarrow$ $2x^2+7x-2x-7 = 0$

$\Rightarrow$ $2x(x-1))+7(x-1) = 0$

$\Rightarrow$ $(x-1)(2x+7) = 0$

$\Rightarrow$ $x = 1 \ or\ x =$ $\frac{(-7)}{2}$

$\\$

Question 15: $6y^2 = 7y+20$

Answer:

$6y^2 = 7y+20$

$\Rightarrow$ $6y^2-7y-20 = 0$

$\Rightarrow$ $6y^2-15y+8y-20 = 0$

$\Rightarrow$ $3y(2y-5)+4(2y-5) = 0$

$\Rightarrow$ $(2y-5)(3y+4) = 0$

$\Rightarrow$ $y =$ $\frac{5}{2}$ $\ or \ y =$ $\frac{(-4)}{3}$

$\\$

Question 16: $4z^2 = 11z+3$

Answer:

$4z^2 = 11z+3$

$\Rightarrow$ $4z^2-11z-3 = 0$

$\Rightarrow$ $4z^2+z-12x-3 = 0$

$\Rightarrow$ $4z(z-3)+(z-3) = 0$

$\Rightarrow$ $(z-3)(4z+1) = 0$

$\Rightarrow$ $z = 3 \ or\ z =$ $\frac{(-1)}{4}$

$\\$

Question 17: $(t-2)^2 = 36$

Answer:

$(t-2)^2 = 36$

$\Rightarrow$ $t^2+4-4t = 36$

$\Rightarrow$ $t^2-4t-32 = 0$

$\Rightarrow$ $t^2-8t+4t-32 = 0$

$\Rightarrow$ $t(t+4)-8(t+4) = 0$

$\Rightarrow$ $(t+4)(t-8) = 0$

$\Rightarrow$ $t = -4 \ or\ t = 8$

$\\$

Question 18: $(2x+3)(x-4) = 6$

Answer:

$(2x+3)(x-4) = 6$

$\Rightarrow$ $2x^2+3x-8x-12 = 6$

$\Rightarrow$ $2x^2-5x-18 = 0$

$\Rightarrow$ $2x^2-9x+4x-18 = 0$

$\Rightarrow$ $2x(x+2)-9(x+2) = 0$

$\Rightarrow$ $2x (x+2)(2x-9) = 0$

$\Rightarrow$ $x = -2 \ or\ x =$ $\frac{9}{2}$

$\\$

Question 19: $12x+7 =$ $\frac{10}{x}$

Answer:

$12x+7 =$ $\frac{10}{x}$

$\Rightarrow$ $12x^2+7x-10 = 0$

$\Rightarrow$ $12x^2+15x-8x-10 = 0$

$\Rightarrow$ $4x(3x-2)+5(3x-2) = 0$

$\Rightarrow$ $(3x-2)(4x+5) = 0$

$\Rightarrow$ $x =$ $\frac{2}{3}$ $\ or \ x =$ $\frac{(-5)}{4}$

$\\$

Question 20: $x^2-10x+21 = 0$

Answer:

$x^2-10x+21 = 0$

$\Rightarrow$ $x^2-7x-3x+21 = 0$

$\Rightarrow$ $x(x-7)-3(x-7) = 0$

$\Rightarrow$ $(x-7)(x-3) = 0$

$\Rightarrow$ $x = 7 \ or \ x = 3$

$\\$

Question 21: $\frac{x}{3}$ $-$ $\frac{6}{x}$ $= 1$

Answer:

$\frac{x}{3}$ $-$ $\frac{6}{x}$ $= 1$

$\Rightarrow$ $x^2-18 = 3x$

$\Rightarrow$ $x^2-3x-18 = 0$

$\Rightarrow$ $x^2-6x+3x-18 = 0$

$\Rightarrow$ $x(x+3)-6(x+3)$

$\Rightarrow$ $(x+3)(x-6) = 0$

$\Rightarrow$ $x = -3 \ or\ x = 6$

$\\$

Question 22: $\frac{(x+1)}{(x+5)}$ $=$ $\frac{(2x-3)}{(x+3)}$

Answer:

$\frac{(x+1)}{(x+5)}$ $=$ $\frac{(2x-3)}{(x+3)}$

$\Rightarrow$ $x^2+x+3x+3 = 2x^2-3x+10x-15$

$\Rightarrow$ $x^2+3x-18 = 0$

$\Rightarrow$ $x^2+6x-3x-18 = 0$

$\Rightarrow$ $x(x-3)+6(x-3) = 0$

$\Rightarrow$ $(x-3)(x+6)$

$\Rightarrow$ $x = 3 \ or\ x = -6$

$\\$

Question 23: $\frac{(x+1)}{(3x-7)}$ $=$ $\frac{(x-1)}{(2x-5)}$

Answer:

$\frac{(x+1)}{(3x-7)}$ $=$ $\frac{(x-1)}{(2x-5)}$

$\Rightarrow$ $2x^2+2x-5x-5 = 3x^2-3x-7x+7$

$\Rightarrow$ $2x^2-3x-5 = 3x^2-10x+7$

$\Rightarrow$ $x^2-7x+12 = 0$

$\Rightarrow$ $x^2-3x-4x+12 = 0$

$\Rightarrow$ $x(x-3)-4(x-3) = 0$

$\Rightarrow$ $(x-3)(x-4) = 0$

$\Rightarrow$ $x = 3 \ or\ x = 4$

$\\$

Question 24: $\frac{1}{(x-1)}$ $-$ $\frac{1}{(x+2)}$ $=$ $\frac{3}{4}$

Answer:

$\frac{1}{(x-1)}$ $-$ $\frac{1}{(x+2)}$ $=$ $\frac{3}{4}$

$\Rightarrow$ $(x+2)-(x-1) =$ $\frac{3}{4}$ $(x-1)(x+2)$

$\Rightarrow$ $3 =$ $\frac{3}{4}$ $(x^2-x+2x-2)$

$\Rightarrow$ $4 = x^2+x-2$

$\Rightarrow$ $x^2+x-6 = 0$

$\Rightarrow$ $x^2+3x-2x-6 = 0$

$\Rightarrow$ $x(x-2)+3(x-2) = 0$

$\Rightarrow$ $(x-2)(x+3) = 0$

$\Rightarrow$ $x = 2 \ or\ x = -3$

$\\$

Question 25: $\frac{4}{(x-1)}$ $-$ $\frac{3}{x}$ $=$ $\frac{5}{x+2}$

Answer:

$\frac{4}{(x-1)}$ $-$ $\frac{3}{x}$ $=$ $\frac{5}{x+2}$

Multiply LHS and RHS by $(x-1)(x)(x+2)$

$\Rightarrow$ $4(x)(x+2)-3(x-1)(x+2) = 5(x-1)(x)$

$\Rightarrow$ $4x^2+8x-3x^2-3x+6 = 5x^2-5x$

$\Rightarrow$ $4x^2+8x-3x^2-3x+6 = 5x^2-5x$

$\Rightarrow$ $x^2+5x+6 = 5x^2-5x$

$\Rightarrow$ $4x^2-10x-6 = 0$

$\Rightarrow$ $2x^2-5x-3 = 0$

$\Rightarrow$ $2x^2-6x+x-3 = 0$

$\Rightarrow$ $2x (x-3)+(x-3) = 1$

$\Rightarrow$ $(x-3)(2x+1) = 0$

$\Rightarrow$ $x = 3 \ or\ -$ $\frac{1}{2}$

$\\$

Question 26: $\frac{(x+2)}{(x-1)}$ $-$ $\frac{4-x}{2x}$ $=$ $\frac{7}{2}$

Answer:

$\frac{(x+2)}{(x-1)}$ $-$ $\frac{4-x}{2x}$ $=$ $\frac{7}{2}$

$\Rightarrow$ $2x(x+2)-(x-1)(4-x) =$ $\frac{7}{2}$ $(x-1)(2x)$

$\Rightarrow$ $2x^2+4x-4x+4+x^2-x = 7(x^2-x)$

$\Rightarrow$ $3x^2-x+4 = 7x^2-7x$

$\Rightarrow$ $4x^2-6x-4 = 0$

$\Rightarrow$ $2x^2-3x-2 = 0$

$\Rightarrow$ $2x^2-4x+x-2 = 0$

$\Rightarrow$ $2x (x-2)+1(x-2) = 0$

$\Rightarrow$ $(x-2)(2x+1) = 0$

$\Rightarrow$ $x = 2 \ or\ x = -$ $\frac{1}{2}$

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