Solving Quadratic Equations with Various Coefficients
The provided content offers solutions to quadratic equations involving different coefficients and variables using LCM calculations. Explore how to solve equations step by step in algebra and enhance your problem-solving skills.
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CHAPTER 04 Algebra III Quadratic Relations Solutions: Practice Questions 4.5
04 Practice Questions 4.5 1. Solve the following equations: 3 x+3 The lowest common denominator of x and 2x is 2x. However, in this case, the solution is easier if we let the LCM be (x)(2x) (i) 2x=2 3 x 3 + = 2 2 x +(x)(2x) (2x)(3)+(x)(3)=(x)(2x)(2) =(x)(2x)(2) (x)(2x)3 3 2x x 6x+3x=(2x2)(2) 9x =4x2 0=4x2-9x 0= x(4x-9) x=0 and 4x-9=0 4x =9 x =9 4
04 Practice Questions 4.5 1. Solve the following equations: 3 4 (ii) 2x+ x+1=5 3 4 x+1=5 LCM =(2x)(x+1) =(2x)(x+1)(5) (x+1)(3)+(2x)(4)=(2x)(x+1)(5) 2x+ +(2x)(x+1) 3 4 (2x)(x+1) x+1 2x 3x+3+8x =(2x2+2x)(5) 3+11x =10x2+10x 0=10x2+10x-11x-3 = 2 0 10 1 3 x x
04 Practice Questions 4.5 1. Solve the following equations: 3 4 (ii) 2x+ x+1=5 10x2-x-3=0 (5x-3)(2x+1)=0 5x-3=0 and 2x+1=0 5x=3 2x=-1 x =3 5 x =-1 2
04 Practice Questions 4.5 1. Solve the following equations: 4a+3 5 2a=11 (iii) 16 The lowest common denominator of 4a, 2a and 16 is 16a. However, in this case, the solution is easier if we let the LCM be (4a)(2a)(16) 5 a 3 a 11 LCM 16 + = = (4 )(2 )(16) a a 4 2 +(4a)(2a)(16) =(4a)(2a)(16)11 5 3 (4a)(2a)(16) 4a (2a)(16)(5)+(4a)(16)(3)=(4a)(2a)(11) 2a 16 (32a)(5)+(64a)(3)=(8a2)(11) 160a+192a=88a2
04 Practice Questions 4.5 1. Solve the following equations: 4a+3 5 2a=11 (iii) 16 160a+192a=88a2 352a=88a2 0=88a2-352a 0=a2-4a 0=a(a-4) a=0 and a-4=0 a=4
04 Practice Questions 4.5 1. Solve the following equations: 3 s- 2 (iv) 2s-1=1 3 s 2 ( )( s ) = = 1 LCM 2 1 s 2 1 s - s ( )2s-1 2s-1 ( = s ( )2s-1 ( 3 s 2 s ( )2s-1 ( ) ( ) ( )1 ( ) ) 2s-1 )3 ( )- s ( )2 ( )= s ( )2s-1 6s-3-2s=2s2-s 2s-3=0 and s-1=0 2s=3 and s=1 s =3 2 4s-3=2s2-s 0=2s2-5s+3 ( )s-1 ( ) 0= 2s-3
04 Practice Questions 4.5 1. Solve the following equations: a-4+1 1 a=2 (v) 3 1 1 a 2 LCM 3 ( ( ( ( )( )( ) a + = = 4 3 a 4 a + a-4 = a-4 1 1 a 2 3 ( )a ( )3 ( ) ( )a ( )3 ( ) )a ( )3 ( ) )a ( )2 ( ) )2 ( ) a-4 a-4 a ( )3 ( )1 ( )+ a-4 ( )3 ( )1 ( )= a-4 3a+3a-12= a2-4a 6a-12=2a2-8a 0=2a2-14a+12 0=a2-7a+6 0= a-6 ( a-6=0 and a-1=0 a=6 a=1 )a-1 ( )
04 Practice Questions 4.5 1. Solve the following equations: 1 x+ x-1=7 3 (vi) 2 1 x+ x-1=7 3 x-1 3 2 LCM = x ( )x-1 ( )2 ( ) + x ( )x-1 x-1 )2 ( )1 ( )+ x ( )2 ( )3 ( )= x ( )x-1 2x-2+6x = x2-x = x ( )x-1 1 x ( 7 2 x ( )x-1 ( )2 ( ) ( )2 ( ) ( ( )2 ( ) )7 ( ) )7 ( ) ( 7x-1=0 and x-2=0 7 1 and x = x =1 7 8x-2=7x2-7x 0=7x2-15x+2 0= 7x-1 ( = 2 x )x-2 ( )
04 Practice Questions 4.5 2. Solve the following equations: 3 4 (i) 2x-1- 3x-1=1 3 2x-1- 4 ( )3x-1 ( )1 ( ) 3x-1=1 LCM= 2x-1 ( - 2x-1 = 2x-1 ( 3 4 ( )3x-1 ( )1 ( ) ( )3x-1 )3 ( )- 2x-1 9x-3-8x+4=6x2-2x-3x+1 x+1=6x2-5x+1 0=6x2-6x 0= x2-x 0= x x-1 ( )1 ( ) ( ( )3x-1 )3x-1 ( ( )1 ( )1 ( ) ) 2x-1 2x-1 3x-1 )4 ( )= 2x-1 3x-1 x =0 and x-1=0 x =1 ( )
04 Practice Questions 4.5 2. Solve the following equations: 6 x-1=7 3 (ii) x-3- 5 6 x-1=7 )5 ( ) x-3 3 ( )x-1 ( )5 ( ) x-3- )x-1 5 LCM= x-3 - x-3 x-1 )5 ( )6 ( )- x-3 5x-5 ( 30x-30-15x+45= x2-4x+3 = x-3 ( ( 6 3 7 5 ( ( ( )x-1 ( )5 ( ) ( )x-1 )x-1 ( ( )5 ( ) )7 ( ) x-3 x-1 )5 ( )3 ( )= x-3 )3 ( )= x2-x-3x+3 ( 15x+15=7x2-28x+21 0=7x2-43x+6 0= 7x-1 ( ( ( )7 ( ) )6 ( )- 5x-15 ( )7 ( ) 7x-1=0 and x-6=0 7 = 1 and x x =1 7 x = 6 )x-6 ( )
04 Practice Questions 4.5 2. Solve the following equations: 7 1 (iii) p+2+ p-1=4 7 1 ( )p-1 ( )1 ( ) p+2+ p-1=4 LCM = p+2 + p+2 p-1 ( = p+2 ( ( ( 7 1 ( )p-1 ( )1 ( ) ( )p-1 )7 ( )+ p+2 7p-7+ p+2= p2- p+2p-2 ( )1 ( ) ( ( )p-1 )p-1 ( ( )1 ( )4 ( ) )4 ( ) )4 ( ) )4 ( ) p+2 p+2 p-1 )1 ( )= p+2 8p-5= p2+ p-2 2p-3=0 and 2p+1=0 2p=3 and 2p=-1 p=3 8p-5=4p2+4p-8 0=4p2-4p-3 0= 2p-3 ( 2 and p=- 1 )2p+1 ( ) 2
04 Practice Questions 4.5 2. Solve the following equations: 8 8 (iv) z-2+ z+2=3 8 8 ( )z+2 ( )1 ( ) z-2+ )z+2 z+2=3 LCM= z-2 )1 ( ) z-2 + z-2 = z-2 ( ( 16z =3z2+6z-6z-12 16z =3z2-12 0=3z2-16z-12 0= 3z+2 ( 8 8 ( ( ( )z+2 )8 ( )+ z-2 8z+16+8z-16= z-2 ( )1 ( ) ( ( )z+2 )z+2 )3z+6 ( ( ( )1 ( )3 ( ) )3 ( ) ) z-2 z+2 )8 ( )= z-2 ( z+2 3z+2=0 and z-6=0 3z=-2 and z=6 z =-2 3 )z-6 ( )
04 Practice Questions 4.5 3. Solve the following equations and leave your answer correct to two decimal places. x+3+4 9 x=1 (i) 6 x+3+4 9 x=1 = x+3 ( ( )x ( )6 ( ) 6 LCM= x+3 + x+3 9 4 x 1 6 ( )x ( )6 ( ) ( )x ( )6 ( ) ( )x ( )6 ( ) x+3 x+3 x ( )6 ( )9 ( )+ x+3 ( )6 ( )4 ( )= x+3 )x ( )1 ( ) 54x+24x+72= x2+3x 78x+72= x2+3x 0= x2-75x-72 a=1, b=-75, c=-72
04 Practice Questions 4.5 3. Solve the following equations and leave your answer correct to two decimal places. x+3+4 9 x=1 (i) x =-b b2-4ac a=1, b=-75, c=-72 6 2a ( ) ( ) ( )( 4 1 ) 2 75 75 2 1 72 = ( ) + 75 5625 2 288 = =75 5913 2 =75 9 73 2 x =75+9 73 =75 95 and x =75-9 73 =-0 95 2 2
04 Practice Questions 4.5 3. Solve the following equations and leave your answer correct to two decimal places. 5 z- z+2=3 4 (ii) 5 5 z 4 + 3 LCM 5 ( )( z )( ) 2 5 = = + z 2 z - z ( )z+2 )5 ( )5 ( )- z ( )5 ( )4 ( )= z ( )z+2 = z ( )z+2 5 z ( 4 3 5 z ( )z+2 ( )5 ( ) ( )5 ( ) ( )5 ( ) z+2 ( )3 ( ) z+2 25z+50-20z =3z2+6z 5z+50=3z2+6z 0=3z2+z-50 = = = 3, 1, 50 a b c
04 Practice Questions 4.5 3. Solve the following equations and leave your answer correct to two decimal places. 5 z- z+2=3 4 (ii) x =-b b2-4ac = = = 3, 1, 50 a b c 5 2a -1 12-4 3 ( )-50 2 3 ( ) -1 1- -600 ( 6 =-1 601 6 ( ) = ) = x =-1+ 601 =3 92 and x =-1- 601 =-4 25 6 6
04 Practice Questions 4.5 3. Solve the following equations and leave your answer correct to two decimal places. 4 7 (iii) m-2+ m+1=3 4 7 m-2+ m+1=3 LCM=(m-2)(m+1)(1) +(m-2)(m+1)(1) =(m-2)(m+1)(1)(3) 4 7 (m-2)(m+1)(1) m-2 m+1 (m+1)(4)+(m-2)(7)=(m-2)(3m+3) 4m+4+7m-14=3m2+3m-6m-6 11m-10=3m2-3m-6 0=3m2-14m+4 3, b = = = 14, 4 a c
04 Practice Questions 4.5 3. Solve the following equations and leave your answer correct to two decimal places. 4 7 (iii) m-2+ m+1=3 = = = 3, 14, 4 a b c x =-b b2-4ac 2a =-(-14) (-14)2-4(3)(4) 2(3) =14 196-48 6 =14 148 6 x =14+ 148 =4 36 and x =14- 148 =0 31 6 6
04 Practice Questions 4.5 3. Solve the following equations and leave your answer correct to two decimal places. 3 4 (iv) 4x+1- x+2=2 3 4 4x+1- x+2=2 LCM=(4x+1)(x+2)(1) -(4x+1)(x+2)(1) =(4x+1)(x+2)(1)(2) 3 4 (4x+1)(x+2)(1) 4x+1 x+2 (x+2)(3)-(4x+1)(4)=(4x+1)(2x+4) 3x+6-16x-4=8x2+16x+2x+4 2-13x =8x2+18x+4 0=8x2+31x+2 a=8, b=31, c=2
04 Practice Questions 4.5 3. Solve the following equations and leave your answer correct to two decimal places. 3 4 (iv) 4x+1- x+2=2 a=8, b=31, c=2 x =-b b2-4ac 2a =-31 (31)2-4(8)(2) 2(8) =-31 961-64 16 =-31 897 16 x =-31 897 =-0 07 and x =-31- 897 =-3 81 16 16
04 Practice Questions 4.5 4. Solve the following equations and leave your answer in surd form. z-4-5 4 (i) z=1 z-4-5 4 z=1 LCM=(z-4)(z)(1) -(z-4)(z)(1)5 =(z-4)(z)(1)(1) 4 (z-4)(z)(1) z-4 z (z)(4)-(z-4)(5)=(z-4)(z) 4z-5z+20= z2-4z 20-z = z2-4z 0= z2-3z-20 a=1, b=-3, c=-20
04 Practice Questions 4.5 4. Solve the following equations and leave your answer in surd form. z-4-5 4 (i) z=1 x =-b b2-4ac a=1, b=-3, c=-20 2a =-(-3) (-3)2-4(1)(-20) 2(1) =3 9-(-80) 2 =3 89 2 x =3+ 89 and x =3- 89 2 2
04 Practice Questions 4.5 4. Solve the following equations and leave your answer in surd form. 3 1 (ii) n+1+ n-1=1 3 1 n+1+ n-1=1 LCM=(n+1)(n-1)(1) +(n+1)(n-1)(1) =(n+1)(n-1)(1)(1) 3 1 (n+1)(n-1)(1) n+1 n-1 (n-1)(3)+(n+1)=(n+1)(n-1) 3n-3+n+1=n2+n-n-1 4n-2=n2-1 0=n2-4n+1 a=1, b=-4, c=1
04 Practice Questions 4.5 4. Solve the following equations and leave your answer in surd form. 3 1 (ii) n+1+ n-1=1 x =-b b2-4ac a=1, b=-4, c=1 2a =-(-4) (-4)2-4(1)(1) 2(1) =4 16-4 2 =4+ 12 2 =2 3 x =2+ 3 and x =2- 3
04 Practice Questions 4.5 4. Solve the following equations and leave your answer in surd form. 6 x-1=7 3 (iii) x+1+ 2 6 x-1=7 3 x+1+ 2 LCM=(x+1)(x-1)(2) +(x+1)(x-1)(2) =(x+1)(x-1)(2)7 6 3 (x+1)(x-1)(2) x+1 x-1 2 (x-1)(2)(6)+(x+1)(2)(3)=(x+1)(7x-7) 12x-12+6x+6=7x2-7x+7x-7 18x-6=7x2-7 0=7x2-18x-1 a=7, b=-18, c=-1
04 Practice Questions 4.5 4. Solve the following equations and leave your answer in surd form. 6 x-1=7 3 (iii) x+1+ x =-b b2-4ac 2 a=7, b=-18, c=-1 2a =-(-18) (-18)2-4(7)(-1) 2(7) =18 324-(-28) 14 =18 352 14 =9 2 22 7 x =9+2 22 7 and x =9-2 22 7
04 Practice Questions 4.5 4. Solve the following equations and leave your answer in surd form. 2 1 (iv) y+1+ y-1= 2 2 1 y+1+ y-1=2 LCM =(y+1)(y-1)(1) +(y+1)(y-1)(1) =(y+1)(y-1)(1)(2) 2 1 (y+1)(y-1)(1) y+1 y-1 (y-1)(2)+(y+1)=(y+1)(2y-2) 2y-2+ y+1=2y2-2y+2y-2 3y-1=2y2-2 0=2y2-3y-1 a=2, b=-3, c=-1
04 Practice Questions 4.5 4. Solve the following equations and leave your answer in surd form. 2 1 (iv) y+1+ y-1= 2 x =-b b2-4ac a=2, b=-3, c=-1 2a =-(-3) (-3)2-4(2)(-1) 2(2) =3 9+8 4 =3 17 4 x =3+ 17 and x =3- 17 4 4
