Question

Please solve with explanation (10 points)

Answer 1

Question (a)

`x^2-a^2-bx-ab=0`

Write in standard form
`ax^2+bx+c=0`:

`\implies x^2-bx-(a^2+ab)=0`

Therefore,

• 
  `a=1`
• 
  `b=-b`
• 
  `c=-(a^2+ab)`

Using quadratic formula:

`x=(-b\pm√(b^2-4ac) )/(2a)`

`\implies x=(b\pm√((-b)^2+4(1)(a^2+ab)) )/(2(1))`

`\implies x=(b\pm√(b^2+4a^2+4ab) )/(2)`

`\implies x=(b\pm√((b+2a)^2) )/(2)`

`\implies x=\frac{b \pm (b+2a)} {2}`

`\implies x=\frac{2b+2a} {2}=b+a`

`\implies x=\frac{-2a} {2}=-a`

Question (b)

`6x^2-15ax=2bx-5ab`

Write in standard form
`ax^2+bx+c=0`:

`\implies 6x^2-15ax-2bx+5ab=0`

`\implies 6x^2-(15a+2b)x+5ab=0`

Therefore,

• 
  `a=6`
• 
  `b=-(15a+2b)`
• 
  `c=5ab`

Using quadratic formula:

`x=(-b\pm√(b^2-4ac) )/(2a)`

`\implies x=((15a+2b)\pm√((-15a-2b)^2-4(6)(5ab)) )/(2(6))`

`\implies x=((15a+2b)\pm√(225a^2+60ab+4b^2-120ab) )/(12)`

`\implies x=((15a+2b)\pm√(225a^2-60ab+4b^2) )/(12)`

`\implies x=((15a+2b)\pm√((15a-2b)^2) )/(12)`

`\implies x=((15a+2b) \pm (15a-2b) )/(12)`

`\implies x=(30a )/(12)=\frac52a`

`\implies x=(4b)/(12)=\frac13b`

Question (c)

`(x^2)/(x-1)=(a^2)/(2a-4)`

Cross multiply:

`x^2(2a-4)=a^2(x-1)`

Write in standard form
`ax^2+bx+c=0`:

`(2a-4)x^2-a^2x+a^2=0`

Therefore,

• 
  `a=(2a-4)`
• 
  `b=-a^2`
• 
  `c=a^2`

Using quadratic formula:

`x=(-b\pm√(b^2-4ac) )/(2a)`

`\implies x=(a^2\pm√((-a^2)^2-4(2a-4)(a^2)) )/(2(2a-4))`

`\implies x=(a^2\pm√(a^4-8a^3+16a^2) )/(4a-8)`

`\implies x=(a^2\pm√(a^2(a-4)^2) )/(4a-8)`

`\implies x=(a^2\pm a(a-4))/(4a-8)`

`\implies x=(a^2\pm (a^2-4a))/(4a-8)`

`\implies x=(2a^2-4a)/(4a-8)=(2a(a-2))/(4(a-2))=\frac12a \ \ \ (a\\eq 2)`

`\implies x=(4a)/(4a-8)=(a)/(a-2) \ \ \ \ (a\\eq 2)`