Quadratics and Cubics

Quadratics and Cubics, SMB-044

Solve for `a` given `8a^3+21=0.`

Round your answer to two decimal places. (2 marks)

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`a=-1.38`

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`8a^3+21`	`=0`
`8a^3`	`=-21`
`a^3`	`=-21/8`
`a`	`=-root3(21/8)`
	`=-1.379…`
	`= -1.38`

Quadratics and Cubics, SMB-043

Solve for `p` given `64p^3+125=0.` (2 marks)

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`p=-5/4`

Show Worked Solution

`64p^3+125`	`=0`
`64p^3`	`=-125`
`p^3`	`=-125/64`
`p`	`=-root3(125/64)`
	`=-(root3(125))/(root3(64))`
	`= -5/4`

Quadratics and Cubics, SMB-042

Solve for `x` given `8x^3=27`. (2 marks)

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`x=3/2`

Show Worked Solution

`8x^3`	`=27`
`x^3`	`=27/8`
`x`	`=root3(27/8)`
	`=(root3(27))/(root3(8))`
	`= 3/2`

Quadratics and Cubics, SMB-041

By completing the square, solve for `b` given

`b^2-10b-125=0.` (3 marks)

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`5+-5sqrt(6)`

Show Worked Solution

`b^2-10b-125`	`=0`
`b^2-10b+25-150`	`=0`
`(b-5)^2`	`=150`
`b-5`	`= +-sqrt(150)`
`b`	`=5+-sqrt(25xx6)`
`b`	`=5+-5sqrt(6)`

Quadratics and Cubics, SMB-040

By completing the square, solve for `x` given

`x^2-12x=-4.` (3 marks)

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`6+-4sqrt(2)`

Show Worked Solution

`x^2-12x`	`=-4`
`x^2-12x+36`	`=-4+36`
`(x-6)^2`	`=32`
`x-6`	`= +-sqrt(32)`
`x`	`=6+-4sqrt(2)`

Quadratics and Cubics, SMB-039

By completing the square, solve for `y` given

`y^2-14y+37=0.` (3 marks)

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`7+-2sqrt(3)`

Show Worked Solution

`y^2-14y+37`	`=0`
`y^2-14y+49-12`	`=0`
`(y-7)^2`	`=12`
`y-7`	`= +-sqrt(12)`
`y`	`=7+-2sqrt(3)`

Quadratics and Cubics, SMB-038

By completing the square, solve for `x` given

`x^2+4x-1 = 0.` (3 marks)

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`-2+-sqrt(5)`

Show Worked Solution

`x^2+4x-1`	`=0`
`x^2+4x+4-5`	`=0`
`(x+2)^2`	`=5`
`x+2`	`= +-sqrt(5)`
`x`	`=-2+-sqrt(5)`

Quadratics and Cubics, SMB-037

Using the quadratic formula, find `p` given

`p^2+2p-4 = 0.` (3 marks)

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`-1 +- sqrt(5)`

Show Worked Solution

`p^2+2p-4 = 0`

`text(Using)\ x = (-b +- sqrt( b^2-4ac) )/(2a)`

`p`	`= (-2 +- sqrt{(2)^2-4 xx 1 xx(-4) })/ (2 xx 1)`
	`= (-2 +- sqrt(20) )/2`
	`=(-2 +- 2sqrt(5) )/2`
	`= -1 +- sqrt(5)`

Quadratics and Cubics, SMB-036

Using the quadratic formula, find `a` given

`5a^2+7a-1 = 0.` (3 marks)

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`(-7 +- sqrt(69) )/10`

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`5a^2+7a-1 = 0`

`text(Using)\ a = (-b +- sqrt( b^2-4ac) )/(2a)`

`a`	`= (-7 +- sqrt{(7)^2-4 xx 5 xx(-1) })/ (2 xx 5)`
	`= (-7 +- sqrt(49+20) )/10`
	`= (-7 +- sqrt(69) )/10`

Quadratics and Cubics, SMB-035

Using the quadratic formula, solve

`3x^2-4x-2 = 0`. (3 marks)

--- 6 WORK AREA LINES (style=lined) ---

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`(2 +- sqrt(10) )/3`

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`3x^2-4x-2 = 0`

`text(Using)\ x = (-b +- sqrt( b^2-4ac) )/(2a)`

`x`	`= (4 +- sqrt{(-4)^2-4 xx 3 xx(-2) })/ (2 xx 3)`
	`= (4 +- sqrt(16+24) )/6`
	`= (4 +- sqrt(40) )/6`
	`= (4 +- 2sqrt(10) )/6`
	`= (2 +- sqrt(10) )/3`

Quadratics and Cubics, SMB-034

Solve the equation `21-4b^2=5b` for `b.` (2 marks)

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`b=7/4 \ text{or}\ -3`

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`21-4b^2`	`=5b`
`4b^2+5b-21`	`=0`
`(4b-7)(b+3)`	`=0`

`4b-7`	`=0`	`text{or}\ \ \ \ b=-3`
`b`	`=7/4`

Quadratics and Cubics, SMB-033

Solve the equation `12a^2+8a-15=0` for `a.` (2 marks)

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`x=5/6 \ text{or}\ -3/2`

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`12a^2+8a-15`	`=0`
`(6a-5)(2a+3)`	`=0`

`6a-5`	`=0`	`text{or}\ \ \ \ a=-3/2`
`a`	`=5/6`

Quadratics and Cubics, SMB-033

Solve the equation `6p^2-p-7=0` for `p`. (2 marks)

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`x=7/6 \ text{or}\ -1`

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`6p^2-p-7`	`=0`
`(6p-7)(p+1)`	`=0`

`6p-7`	`=0`	`text{or}\ \ \ \ p=-1`
`p`	`=7/6`

Quadratics and Cubics, SMB-032

Solve the equation `6x^2-3x-9=0` for `x`. (2 marks)

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`x=3/2 \ text{or}\ -1`

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`6x^2-3x-9`	`=0`
`3(2x^2-x-3)`	`=0`
`3(2x-3)(x+1)`	`=0`

`2x-3`	`=0`	`text{or}\ \ \ \ x=-1`
`x`	`=3/2`

Quadratics and Cubics, SMB-031

Solve the equation `3q^2-10q-8=0` for `q.` (2 marks)

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`q=-2/3 \ text{or}\ 4`

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`3q^2-10q-8`	`=0`
`(3q+2)(q-4)`	`=0`

`3q+2`	`=0`	`text{or}\ \ q=4`
`q`	`=-2/3`

Quadratics and Cubics, SMB-030

Solve the equation `p^2-12p=64` for `p`. (2 marks)

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`p=16 \ text{or}\ -4`

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`p^2-12p`	`=64`
`p^2-12p-64`	`=0`
`(p-16)(p+4)`	`=0`

`:. p=16 \ text{or}\ -4`

Quadratics and Cubics, SMB-029

Solve the equation `14x=32-x^2` for `x`. (2 marks)

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`x=2 \ text{or}\ -16`

Show Worked Solution

`14x`	`=32-x^2`
`x^2+14x-32`	`=0`
`(x-2)(x+16)`	`=0`

`:. x=2 \ text{or}\ -16`

Quadratics and Cubics, SMB-028

Solve the equation `c^2-24=5c` for `c`. (2 marks)

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`c=8 \ text{or}\ -3`

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`c^2-24`	`=5c`
`c^2-5c-24`	`=0`
`(c-8)(c+3)`	`=0`

`:. c=8 \ text{or}\ -3`

Quadratics and Cubics, SMB-027

Solve the equation `y^2-2y-3=0` for `y`. (2 marks)

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`y=3 \ text{or}\ -1`

Show Worked Solution

`y^2-2y-3`	`=0`
`(y-3)(y+1)`	`=0`

`:. y=3 \ text{or}\ -1`

Quadratics and Cubics, SMB-025

Solve the equation `t^2-8t+12=0` for `t`. (2 marks)

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`:. t=6 \ text{or}\ 2`

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`t^2-8t+12`	`=0`
`(t-6)(t-2)`	`=0`

`:. t=6 \ text{or}\ 2`

Non-Linear, SMB-024

The volume of a sphere is given by `V = 4/3 pi r^3` where `r` is the radius of the sphere.

If the volume of a sphere is `220\ text(cm)^3`, find the radius, to 1 decimal place. (3 marks)

--- 6 WORK AREA LINES (style=lined) ---

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`3.7\ \ text{cm (to 1 d.p.)}`

Show Worked Solution

`V`	`= 4/3 pi r^3`
`3V`	`= 4 pi r^3`
`r^3`	`= (3V)/(4 pi)`

`text(When)\ \ V = 220`

`r^3`	`= (3 xx 220)/(4 pi)`
	`= 52.521…`
`:. r`	`=root3 (52.521…)`
	`= 3.744…\ \ \ text{(by calc)}`
	`= 3.7\ \ text{cm (to 1 d.p.)}`

Non-Linear, SMB-023

Make `r` the subject of the equation `V = 4/3 pir^3`. (3 marks)

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`r = root(3)((3V)/(4pi))`

Show Worked Solution

`V`	`= 4/3 pir^3`
`3V`	`=4pir^3`
`(3V)/4`	`= pir^3`
`r^3`	`= (3V)/(4pi)`
`r`	`= root(3)((3V)/(4pi))`

Functions, 2ADV F1 2017 HSC 2 MC

Which expression is equal to `3x^2-x-2`?

`(3x-1) (x + 2)`
`(3x + 1) (x-2)`
`(3x-2) (x + 1)`
`(3x + 2) (x-1)`

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`D`

Show Worked Solution

`3x^2-x-2= (3x + 2) (x-1)`

`=> D`

Algebra, STD2 A1 2005 HSC 24c

Make `L` the subject of the equation `T = 2piL^2`. (2 marks)

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`± sqrt(T/(2pi))`

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`T`	`= 2piL^2`
`L^2`	`= T/(2pi)`
`:.L`	`= ±sqrt(T/(2pi))`

Algebra, STD2 A1 2004 HSC 11 MC

If `d = 6t^2`, what is a possible value of `t` when `d = 2400`?

`0.05`
`20`
`120`
`400`

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`B`

Show Worked Solution

`d`	`= 6t^2`
`t^2`	`= d/6`
`t`	`= +- sqrt(d/6)`

`text(When)\ \ d = 2400:`

`t= +- sqrt(2400/6)= +- 20`

`=> B`

Functions, 2ADV F1 2010 HSC 1a

Solve `x^2 = 4x`. (2 marks)

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`x = 0\ text(or)\ 4`

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

`:.\ x = 0\ text(or)\ 4`

Functions, 2ADV F1 2013 HSC 1 MC

What are the solutions of `2x^2-5x-1 = 0`?

`x = (-5 +-sqrt17)/4`
`x = (5 +-sqrt17)/4`
`x = (-5 +-sqrt33)/4`
`x = (5 +-sqrt33)/4`

Show Answers Only

`D`

Show Worked Solution

`2x^2-5x-1 = 0`

`text(Using)\ x = (-b +- sqrt( b^2-4ac) )/(2a)`

`x`	`= (5 +- sqrt{\ \ (-5)^2-4 xx 2 xx(-1) })/ (2 xx 2)`
	`= (5 +- sqrt(25 + 8) )/4`
	`= (5 +- sqrt(33) )/4`

`=> D`

Algebra, STD2 A1 2011 HSC 18 MC

Which of the following correctly expresses `a` as the subject of `s= ut+1/2at^2 `?

`a=(2(s-ut))/t^2`
`a=(2s-ut)/t^2`
`a=(1/2(s-ut))/t^2`
`a=(1/2s-ut)/t^2`

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`A`

Show Worked Solution

`s`	`=ut+1/2at^2`
`1/2at^2`	`=s-ut`
`at^2`	`=2(s-ut)`
`a`	`=(2(s-ut))/t^2`

`=>A`