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Equations That May be Reduced to Quadratics=
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In this section ,we solve equations that may be reduced to <= /span>2<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>nd degree equations by substitution m= ethod
Equations That May be Reduced to Quadratics=
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In this section ,we solve equations that may be reduced to <= /span>2<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>nd degree equations by substitution m= ethod and fractional equations that reduce to 2nd = degree equations
Equations That May be Reduced to Quadratics=
------=_NextPart_01C9D798.C11EECC0 Content-Location: file:///C:/EA4BB0D3/5EquationsThatMaybeReducedtoQuadratics_files/slide0074.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Equations
In this section ,we solve equations that may be reduced to <= /span>2<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>nd degree equations by substitution m= ethod and fractional equations that reduce to 2nd = degree equations
Equations That May be Reduced to Quadratics=
Substitution Method=
If a pattern is repeat= ed many times in an expression, we may substitute the pattern= with a variable to make the equation look simpler. <= /span>
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In this section ,we solve equations that may be reduced to <= /span>2<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>nd degree equations by substitution m= ethod and fractional equations that reduce to 2nd = degree equations
Equations That May be Reduced to Quadratics=
Substitution Method=
If a pattern is repeat= ed many times in an expression, we may substitute the pattern= with a variable to make the equation look simpler. <= /span>
Example = A.
 
a. Given=
(
x
x – 1
)
2
–
(
x
x – 1
)
– 6
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In this section ,we solve equations that may be reduced to <= /span>2<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>nd degree equations by substitution m= ethod and fractional equations that reduce to 2nd = degree equations
Equations That May be Reduced to Quadratics=
Substitution Method=
If a pattern is repeat= ed many times in an expression, we may substitute the pattern= with a variable to make the equation look simpler. <= /span>
Example = A.
 
a. Given=
 
if we substitute y for
(
x
x – 1
)
2
–
(
x
x – 1
)
– 6
x
x – 1
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In this section ,we solve equations that may be reduced to <= /span>2<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>nd degree equations by substitution m= ethod and fractional equations that reduce to 2nd = degree equations
Equations That May be Reduced to Quadratics=
Substitution Method=
If a pattern is repeat= ed many times in an expression, we may substitute the pattern= with a variable to make the equation look simpler. <= /span>
Example = A.
 
a. Given=
 
if we substitute y for
then the expression is y– y – 6.
(
x
x – 1
)
2
–
(
x
x – 1
)
– 6
x
x – 1
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In this section ,we solve equations that may be reduced to <= /span>2<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>nd degree equations by substitution m= ethod and fractional equations that reduce to 2nd = degree equations
Equations That May be Reduced to Quadratics=
Substitution Method=
If a pattern is repeat= ed many times in an expression, we may substitute the pattern= with a variable to make the equation look simpler. <= /span>
Example = A.
 
a. Given=
 
if we substitute y for
then the expression is y– y – 6.
(
x
x – 1
)
2
–
(
x
x – 1
)
– 6
x
x – 1
b. Given (x2 – 1)2 – 3(x2 – 1) + 2
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In this section ,we solve equations that may be reduced to <= /span>2<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>nd degree equations by substitution m= ethod and fractional equations that reduce to 2nd = degree equations
Equations That May be Reduced to Quadratics=
Substitution Method=
If a pattern is repeat= ed many times in an expression, we may substitute the pattern= with a variable to make the equation look simpler. <= /span>
Example = A.
 
a. Given=
 
if we substitute y for
then the expression is y– y – 6.
(
x
x – 1
)
2
–
(
x
x – 1
)
– 6
x
x – 1
b. Given= (x2 – = 1)2 – 3(x2 – 1) + 2
 
if we substitute y for (x – 1)
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In this section ,we solve equations that may be reduced to <= /span>2<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>nd degree equations by substitution m= ethod and fractional equations that reduce to 2nd = degree equations
Equations That May be Reduced to Quadratics=
Substitution Method=
If a pattern is repeat= ed many times in an expression, we may substitute the pattern= with a variable to make the equation look simpler. <= /span>
Example = A.
 
a. Given=
 
if we substitute y for
then the expression is y– y – 6.
(
x
x – 1
)
2
–
(
x
x – 1
)
– 6
x
x – 1
b. Given= (x2 – = 1)2 – 3(x2 – 1) + 2
 
if we substitute y for (x – 1)
then the expression is y– 3y + 2.
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c. Given x4 – 5x2 – 14,
Equations That May be Reduced to Quadratics=
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c. Given= x4 – 5x2 – 14,
if we substitute y =3D x so x4 =3D y2,
Equations That May be Reduced to Quadratics=
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c. Given= x4 – 5x2 – 14,
if we substitute y =3D x so x4 =3D y2,
then the expression is y– 5y – 14.
Equations That May be Reduced to Quadratics=
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c. Given= x4 – 5x2 – 14,
if we substitute y =3D x so x4 =3D y2,
then the expression is y– 5y – 14.
Equations That May be Reduced to Quadratics=
To solve an equation via substitution, instead of solving one difficult equation, we solve two easy equations.
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c. Given= x4 – 5x2 – 14,
if we substitute y =3D x so x4 =3D y2,
then the expression is y– 5y – 14.
Equations That May be Reduced to Quadratics=
To solve an equation via substitution, instead of solving one difficult equation, we solve two easy equations. We first solve the 2nd degree equations obtained after the substitution,
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c. Given= x4 – 5x2 – 14,
if we substitute y =3D x so x4 =3D y2,
then the expression is y– 5y – 14.
Equations That May be Reduced to Quadratics=
To solve an equation via substitution, instead of solving one difficult equation, we solve two easy equations. We first solve the 2nd degree equations obtained after the substitution, than put the answers back to the substituted pattern and solve those equations.
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c. Given= x4 – 5x2 – 14,
if we substitute y =3D x so x4 =3D y2,
then the expression is y– 5y – 14.
Equations That May be Reduced to Quadratics=
To solve an equation via substitution, instead of solving one difficult equation, we solve two easy equations. We first solve the 2nd degree equations obtained after the substitution, than put the answers back to the substituted pattern and solve those equations.
Example B.  Solve x4 – 5x2 – 14 =3D 0 for x.
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c. Given= x4 – 5x2 – 14,
if we substitute y =3D x so x4 =3D y2,
then the expression is y– 5y – 14.
Equations That May be Reduced to Quadratics=
To solve an equation via substitution, instead of solving one difficult equation, we solve two easy equations. We first solve the 2nd degree equations obtained after the substitution, than put the answers back to the substituted pattern and solve those equations.
Example B.  Solve x4 – 5x2 – 14 =3D 0 for x.
We subst= itute y =3D x2 so x4 =3D y2, the equation is
        = ;     y2 – 5y – 14 =3D 0    (1st equation to solve)
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c. Given= x4 – 5x2 – 14,
if we substitute y =3D x so x4 =3D y2,
then the expression is y– 5y – 14.
Equations That May be Reduced to Quadratics=
To solve an equation via substitution, instead of solving one difficult equation, we solve two easy equations. We first solve the 2nd degree equations obtained after the substitution, than put the answers back to the substituted pattern and solve those equations.
Example B.  Solve x4 – 5x2 – 14 =3D 0 for x.
We subst= itute y =3D x2 so x4 =3D y2, the equation is
        = ;     y2 – 5y – 14 =3D 0    (1st equation to solve)
(y – 7)(y + 2) =3D 0&= #13;
y =3D 7, y =3D –2
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c. Given= x4 – 5x2 – 14,
if we substitute y =3D x so x4 =3D y2,
then the expression is y– 5y – 14.
Equations That May be Reduced to Quadratics=
To solve an equation via substitution, instead of solving one difficult equation, we solve two easy equations. We first solve the 2nd degree equations obtained after the substitution, than put the answers back to the substituted pattern and solve those equations.
Example B.  Solve x4 – 5x2 – 14 =3D 0 for x.
We subst= itute y =3D x2 so x4 =3D y2, the equation is
        = ;     y2 – 5y – 14 =3D 0    (1st equation to solve)
(y – 7)(y + 2) =3D 0&= #13;
y =3D 7, y =3D –2
To find = x, since y =3D x, so 7 =3D x2  &= nbsp; and   –2 =3D x2
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c. Given= x4 – 5x2 – 14,
if we substitute y =3D x so x4 =3D y2,
then the expression is y– 5y – 14.
Equations That May be Reduced to Quadratics=
To solve an equation via substitution, instead of solving one difficult equation, we solve two easy equations. We first solve the 2nd degree equations obtained after the substitution, than put the answers back to the substituted pattern and solve those equations.
Example B.  Solve x4 – 5x2 – 14 =3D 0 for x.
We subst= itute y =3D x2 so x4 =3D y2, the equation is
        = ;     y2 – 5y – 14 =3D 0    (1st equation to solve)
(y – 7)(y + 2) =3D 0&= #13;
y =3D 7, y =3D –2
To find = x, since y =3D x, so 7 =3D x2  &= nbsp; and   –2 =3D x2    
(2nd equation to solve)  
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c. Given= x4 – 5x2 – 14,
if we substitute y =3D x so x4 =3D y2,
then the expression is y– 5y – 14.
Equations That May be Reduced to Quadratics=
To solve an equation via substitution, instead of solving one difficult equation, we solve two easy equations. We first solve the 2nd degree equations obtained after the substitution, than put the answers back to the substituted pattern and solve those equations.
Example B.  Solve x4 – 5x2 – 14 =3D 0 for x.
We subst= itute y =3D x2 so x4 =3D y2, the equation is
        = ;     y2 – 5y – 14 =3D 0    (1st equation to solve)
(y – 7)(y + 2) =3D 0&= #13;
y =3D 7, y =3D –2
To find = x, since y =3D x, so 7 =3D x2  &= nbsp; and   –2 =3D x2    
(2nd equation to solve)   ±Ö7 =3D x        = ;   ±iÖ2 =3D x
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Example C.  Solve        = ;            &n= bsp;            = ;            &n= bsp; for x.
(
x
x – 1
)
2
–
(
x
x – 1
)
– 6 =3D 0
Equations That May be Reduced to Quadratics=
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Example C.  Solve        = ;            &n= bsp;            = ;            &n= bsp; for x.
We subst= itute y =3D        = ;     the equation is
 
y= 2 – y – 6 =3D 0    = (1st equation to solve)
(
x
x – 1
)
2
–
(
x
x – 1
)
– 6 =3D 0
x
x – 1
Equations That May be Reduced to Quadratics=
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Example C.  Solve        = ;            &n= bsp;            = ;            &n= bsp; for x.
We subst= itute y =3D        = ;     the equation is
 
y= 2 – y – 6 =3D 0    = (1st equation to solve)
(y ̵= 1; 3)(y + 2) =3D 0
y =3D 3,= y =3D –2
(
x
x – 1
)
2
–
(
x
x – 1
)
– 6 =3D 0
x
x – 1
Equations That May be Reduced to Quadratics=
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Example C.  Solve        = ;            &n= bsp;            = ;            &n= bsp; for x.
We subst= itute y =3D        = ;     the equation is
 
y= 2 – y – 6 =3D 0    = (1st equation to solve)
(y ̵= 1; 3)(y + 2) =3D 0
y =3D 3, y =3D –2
To find = x, since y =3D
        = ;  
so 3 =3D        = ;         and  –= 2 =3D        = ;          (2 equation to solve)
(
x
x – 1
)
2
–
(
x
x – 1
)
– 6 =3D 0
x
x – 1
x
x – 1
x
x – 1
x
x – 1
Equations That May be Reduced to Quadratics=
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Example C.  Solve        =             &nb= sp;            =             &nb= sp; for x.
We subst= itute y =3D        = ;     the equation is
 
y= 2 – y – 6 =3D 0    = (1st equation to solve)
(y ̵= 1; 3)(y + 2) =3D 0
y =3D 3, y =3D –2
To find = x, since y =3D
        = ;  
so 3 =3D        = ;         and  –= 2 =3D        = ;          (2 equation to solve)
3(x – 1) =3D x    &= nbsp;
(
x
x – 1
)
2
–
(
x
x – 1
)
– 6 =3D 0
x
x – 1
x
x – 1
x
x – 1
x
x – 1
Equations That May be Reduced to Quadratics=
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Example C.  Solve        = ;            &n= bsp;            = ;            &n= bsp; for x.
We subst= itute y =3D        = ;     the equation is
 
y= 2 – y – 6 =3D 0    = (1st equation to solve)
(y ̵= 1; 3)(y + 2) =3D 0
y =3D 3, y =3D –2
To find = x, since y =3D
        = ;  
so 3 =3D        = ;         and  –= 2 =3D        = ;          (2 equation to solve)
3(x – 1) =3D x       = 3;
3x – 3 =3D x        = ;   
   
(
x
x – 1
)
2
–
(
x
x – 1
)
– 6 =3D 0
x
x – 1
x
x – 1
x
x – 1
x
x – 1
Equations That May be Reduced to Quadratics=
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Example C.  Solve        = ;            &n= bsp;            = ;            &n= bsp; for x.
We subst= itute y =3D        = ;     the equation is
 
y= 2 – y – 6 =3D 0    = (1st equation to solve)
(y ̵= 1; 3)(y + 2) =3D 0
y =3D 3, y =3D –2
To find = x, since y =3D
        = ;  
so 3 =3D        = ;         and  –= 2 =3D        = ;          (2 equation to solve)
3(x – 1) =3D x       = 3;
3x – 3 =3D x        = ;   
        x =3D 3/2        = ;       
(
x
x – 1
)
2
–
(
x
x – 1
)
– 6 =3D 0
x
x – 1
x
x – 1
x
x – 1
x
x – 1
Equations That May be Reduced to Quadratics=
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Example C.  Solve        = ;            &n= bsp;            = ;            &n= bsp; for x.
We subst= itute y =3D        = ;     the equation is
 
y= 2 – y – 6 =3D 0    = (1st equation to solve)
(y ̵= 1; 3)(y + 2) =3D 0
y =3D 3, y =3D –2
To find = x, since y =3D
        = ;  
so 3 =3D        = ;         and  –= 2 =3D        = ;          (2 equation to solve)
3(x – 1) =3D x       –2(x – 1) =3D x
3x – 3 =3D x        = ;   
        x =3D 3/2        = ;       
(
x
x – 1
)
2
–
(
x
x – 1
)
– 6 =3D 0
x
x – 1
x
x – 1
x
x – 1
x
x – 1
Equations That May be Reduced to Quadratics=
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Example C.  Solve        = ;            &n= bsp;            = ;            &n= bsp; for x.
We subst= itute y =3D        = ;     the equation is
 
y= 2 – y – 6 =3D 0    = (1st equation to solve)
(y ̵= 1; 3)(y + 2) =3D 0
y =3D 3, y =3D –2
To find = x, since y =3D
        = ;  
so 3 =3D        = ;         and  –= 2 =3D        = ;          (2 equation to solve)
3(x – 1) =3D x       –2(x – 1) =3D x
3x – 3 =3D x        = ;    –2x + 2 =3D x
        x =3D 3/2        = ;       
(
x
x – 1
)
2
–
(
x
x – 1
)
– 6 =3D 0
x
x – 1
x
x – 1
x
x – 1
x
x – 1
Equations That May be Reduced to Quadratics=
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Example C.  Solve        = ;            &n= bsp;            = ;            &n= bsp; for x.
We subst= itute y =3D        = ;     the equation is
 
y= 2 – y – 6 =3D 0    = (1st equation to solve)
(y ̵= 1; 3)(y + 2) =3D 0
y =3D 3, y =3D –2
To find = x, since y =3D
        = ;  
so 3 =3D        = ;         and  –= 2 =3D        = ;          (2 equation to solve)
3(x – 1) =3D x       –2(x – 1) =3D x
3x – 3 =3D x        = ;    –2x + 2 =3D x
        x =3D 3/2        = ;        2/3 =3D x
(
x
x – 1
)
2
–
(
x
x – 1
)
– 6 =3D 0
x
x – 1
x
x – 1
x
x – 1
x
x – 1
Equations That May be Reduced to Quadratics=
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Example D.  Solve 2x2/3 + x1/3 – 6 =3D 0 for x.
Equations That May be Reduced to Quadratics=
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Example D.  Solve 2x2/3 + x1/3 – 6 =3D 0 for x. =
We subst= itute y =3Dx1/3 
Equations That May be Reduced to Quadratics=
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Example D.  Solve 2x2/3 + x1/3 – 6 =3D 0 for x. =
We subst= itute y =3Dx1/3 so x2/3 =3D y2,
Equations That May be Reduced to Quadratics=
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Example D.  Solve 2x2/3 + x1/3 – 6 =3D 0 for x. =
We subst= itute y =3Dx1/3 so x2/3 =3D y2, the equation is
        = ;   2y2 + y – 6 =3D 0        = ;   (1 equation to solve)
Equations That May be Reduced to Quadratics=
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