MARC:
Maths question
MARC:
\(36x^2=100\)
MARC:
Divide both sides by 36
MARC:
\(\frac{36x^{2}}{36}=\frac{100}{36}\)
MARC:
\(x^2=\frac{100}{36}\)
MARC:
simplify it...
MARC:
\(x^2=\frac{25\times4}{9\times4}\)
MARC:
\(x^{2}=\frac{25}{9}\times\frac{4}{4}\)
MARC:
\(x^{2}=\frac{25}{9}\)
MARC:
square root both sides...
MARC:
\(\sqrt{x^{2}}=\pm\sqrt{\frac{25}{9}}\)
MARC:
\(x=\frac{5}{3},-\frac{5}{3}\)
MARC:
Therefore,answer is D.
MARC:
http://assets.openstudy.com/updates/attachments/584a07d1e4b0ff7d2a1cd1be-marc_d-1481281368620-screenshot_137.png http://assets.openstudy.com/updates/attachments/584a07d1e4b0ff7d2a1cd1be-marc_d-1481281636027-screenshot_138.png http://assets.openstudy.com/updates/attachments/584a07d1e4b0ff7d2a1cd1be-marc_d-1481281635406-screenshot_140.png http://assets.openstudy.com/updates/attachments/584a07d1e4b0ff7d2a1cd1be-marc_d-1481281635146-screenshot_141.png http://assets.openstudy.com/updates/attachments/584a07d1e4b0ff7d2a1cd1be-marc_d-1481281634796-screenshot_142.png http://assets.openstudy.com/updates/attachments/584a07d1e4b0ff7d2a1cd1be-marc_d-1481281634591-screenshot_143.png http://assets.openstudy.com/updates/attachments/584a07d1e4b0ff7d2a1cd1be-marc_d-1481281633893-screenshot_144.png http://assets.openstudy.com/updates/attachments/584a07d1e4b0ff7d2a1cd1be-marc_d-1481281633609-screenshot_145.png http://assets.openstudy.com/updates/attachments/584a07d1e4b0ff7d2a1cd1be-marc_d-1481281633290-screenshot_146.png http://assets.openstudy.com/updates/attachments/584a07d1e4b0ff7d2a1cd1be-marc_d-1481281632981-screenshot_147.png http://assets.openstudy.com/updates/attachments/584a07d1e4b0ff7d2a1cd1be-marc_d-1481281632687-screenshot_148.png http://assets.openstudy.com/updates/attachments/584a07d1e4b0ff7d2a1cd1be-marc_d-1481281632392-screenshot_149.png http://assets.openstudy.com/updates/attachments/584a07d1e4b0ff7d2a1cd1be-marc_d-1481281631022-screenshot_150.png http://assets.openstudy.com/updates/attachments/584a07d1e4b0ff7d2a1cd1be-marc_d-1481281630753-screenshot_151.png http://assets.openstudy.com/updates/attachments/584a07d1e4b0ff7d2a1cd1be-marc_d-1481281630507-screenshot_152.png http://assets.openstudy.com/updates/attachments/584a07d1e4b0ff7d2a1cd1be-marc_d-1481281630153-screenshot_153.png
MARC:
...
MARC:
\(x^2+12x+36=25\)
MARC:
make the eqn into general form ax^2+bx+c=0
MARC:
subtract both sides by 25
MARC:
\(x^{2}+12x+36-25=0\)
MARC:
\(x^{2}+12x+11=0\)
MARC:
Factorise the quadractic eqn...
MARC:
|dw:1481282101895:dw|
MARC:
\((x+11)(x+1)=0\)
MARC:
u can double check ur answer by expanding the bracket... \((x+11)(x+1)=0\)
MARC:
solve for x
MARC:
\(x+11=0\) \(x+1=0\) \(x=-11\) \(x=-1\)
MARC:
Therefore,answer is C.
MARC:
...
MARC:
The vertex form of a quadratic function is given by. f (x) = a(x - h)2 + k, where (h, k) is the vertex of the parabola. FYI: Different textbooks have different interpretations of the reference "standard form" of a quadratic function.
MARC:
\(y=\frac{3}{5}x^{2}+30x+382\)
MARC:
Factorise 3/5 on the right hand side eqn...
MARC:
\(y=\frac{3}{5}(x^{2}+50x+\frac{1910}{3})\)
MARC:
\(y=\frac{3}{5}(x^{2}+50x+(\frac{50}{2})^{2}-(\frac{50}{2})^{2}+\frac{1910}{3})\)
MARC:
\(y=\frac{3}{5}(x^{2}+(25)^{2}-(25)^{2}+\frac{1910}{3})\)
MARC:
\(y=\frac{3}{5}[(x+25)^{2}-625+\frac{1910}{3}]\)
MARC:
\(y=\frac{3}{5}[(x+25)^{2}+\frac{35}{3}]\)
MARC:
\(y=\frac{3}{5}(x+25)^{2}+7\)
MARC:
To get a vertex form, u must use completing the square method as shown above...
MARC:
Now,u can find the vertex and the y-intercept...
MARC:
\(x+25=0\) \(x=-25\) Therefore,the minimum point is (-25,7)
MARC:
Thus,the vertex is =(-25,7)
MARC:
when x=0, \(y=\frac{3}{5}(0+25)^{2}+7\)
MARC:
\(y=375+7\)
MARC:
\(y=382\)
MARC:
Thus,the y-intercept is 382.
MARC:
Therefore,the answer is A.
MARC:
....
MARC:
\(x^{2}+18x+(~~~)\)Fill in the blank.
MARC:
First,list out the factors of 18 which are 1,2,3,6,9 and 18.
MARC:
|dw:1481290755761:dw|
MARC:
Think:what are the two values u needed when u multiply both numbers in order to get 18?
MARC:
The possible answer are 9 and 2 or 3 and 6...
MARC:
|dw:1481291364647:dw|
MARC:
|dw:1481291595076:dw|
MARC:
what value u get when u multiply 9 with 2?
MARC:
Answer is 18.
MARC:
Do the samething for 3 and 6.
MARC:
what value u get when u multiply 3 with 6?
MARC:
Answer is also 18.
MARC:
\(x^{2}+18x+(\color{red}{18})\)
MARC:
Therefore,answer is B.
MARC:
...
MARC:
\(x^{2}-x+(~~~)\) Fill in the blank.
MARC:
Do the samething as question 1.
MARC:
The pattern is similar...
MARC:
\(x^{2}\color{red}{-1}x+(~~~)\)
MARC:
First,list the factors of 1.
MARC:
Factors of 1 is 1.
MARC:
Think:what are the two values u needed when u multiply both numbers in order to get -1?
MARC:
Answer is -1 and 1.
MARC:
what value u get when u multiply -1 with -1?
MARC:
Answer is 1.
MARC:
\(x^{2}-x+(\color{red}{1})\)
MARC:
Therefore,answer is A
MARC:
...
MARC:
\(x^{2}-24x+(~~~)\) Fill in the blank.
MARC:
Samething here...
MARC:
similar to question 1 and 2.
MARC:
Pattern similar*
MARC:
\(x^{2}\color{red}{-24}x+(~~~)\)
MARC:
Start by listing the factors of 24.
MARC:
Factors of 24: 1,2,3,4,6,8,12,24
MARC:
....
MARC:
\(3x^{2}+x=\frac{2}{3}\) Find the values of x.
MARC:
make the eqn into the general form \(ax^{2}+bx+c=0\)
MARC:
\(3x^{2}+x-\frac{2}{3}=0\)
MARC:
we do not like fractions when factoring quadratic eqn...
MARC:
multiply the eqn by 3
MARC:
\(\color{red}{3}\times(3x^{2}+x-\frac{2}{3}=0)\)
MARC:
\(9x^{2}+3x-2=0\)
MARC:
Factorise the quadratic eqn.
MARC:
|dw:1481356581737:dw|
MARC:
u can double check ur quadratic eqn by opening the bracket...
MARC:
\(\color{red}{(}3x+2\color{red}{)}\color{red}{(}3x-1\color{red}{)}=0\)
MARC:
solve for x...
MARC:
\(3x+2=0\) \(x=-\frac{2}{3}\)
MARC:
\(3x-1=0\) \(x=\frac{1}{3}\)
MARC:
Thus,the values of x are -2\3 and 1\3
MARC:
Therefore,answer is B.
MARC:
...
MARC:
Solve the eqn. \(x^{2}-30x+225=400\)
MARC:
Make the eqn into its general form \(ax^{2}+bx+c=0\)
MARC:
subtract both sides by 400
MARC:
\(x^{2}-30x+225-400=400-400\)
MARC:
\(x^{2}-30x-175=0\)
MARC:
Factorise the quadratic eqn.
MARC:
similar as question \(3x^{2}+x=\frac{2}{3}\)
MARC:
now,let's try a new method by using a calculator
MARC:
i can't teach u how to use a calculator bcoz we might not hv the same calculator...
MARC:
u can search the manual instructions in the internet.
MARC:
\((x-35)(x+5)=0\)
MARC:
solve for x.
MARC:
\(x-35=0\) \(x=35\)
MARC:
\(x+5=0\) \(x=-5\)
MARC:
Thus,the values of x are 35 and -5
MARC:
Therefore,the answer is B
MARC:
...
MARC:
4. Find the dimensions of the garden with the maximum area.
MARC:
Perimeter of a fence for fencing around the rectangular garden=100 feet
MARC:
\(Area=50x-x^{2}\)
MARC:
First,sketch a rectangular garden.
MARC:
|dw:1481358644570:dw|
MARC:
Let say x=length and y=width
MARC:
\(2x+2y=100\)
MARC:
Factorise 2 on the left hand side
MARC:
\(\color{red}{2}(x+y)=100\)
MARC:
divide both sides by 2
MARC:
\(x+y=50\)
MARC:
\(x=50-y\)
MARC:
Substitute \(x=50-y\) into \(A=50x-x^{2}\)
MARC:
\(A=50(50-y)-(50-y)^{2}\)
MARC:
Expand the brackets...
MARC:
\(A=2500-50y-[(50-y)(50-y)]\)
MARC:
\(A=2500-50y-(2500+y^2-50y-50y)\)
MARC:
\(A=2500-50y-(2500+y^{2}-100y)\)
MARC:
\(A=2500-50y-2500-y^{2}+100y\)
MARC:
\(A=-y^{2}+50y\)
MARC:
Differentiate A...
MARC:
\(\frac{dA}{dx}=-2y+50\)
MARC:
When area is maximum,\(\frac{dA}{dx}=0\),
MARC:
\(0=-2y+50\)
MARC:
\(2y=50\) \(y=25\)
MARC:
when \(y=25\),
MARC:
\(x+y=50\)
MARC:
\(x+25=50\)
MARC:
\(x=25\)
MARC:
Therefore,the length is 25 and the width is also 25.
MARC:
Answer is B.
MARC:
...
MARC:
Solve the eqn. \(x^2-4x+4=100\)
MARC:
Make it into its general form \(ax^{2}+bx+c=0\)
MARC:
\(x^2-4x+4-100=0\)
MARC:
\(x^2-4x-96=0\)
MARC:
Next,factorise the quadratic eqn.
MARC:
|dw:1481602220858:dw|
MARC:
\((x-12)(x+8)=0\)
MARC:
u can double check ur eqn by expanding the brackets.
MARC:
\(\color{red}{(}x-12\color{red}{)}\color{red}{(}x+8\color{red}{)}=0\)
MARC:
okay,solve for x.
MARC:
\(x-12=0\) x=12
MARC:
\(x+8=0\) \(x=-8\)
MARC:
Thus,the values of x are -8 and 12.
MARC:
Therefore,answer is B.
MARC:
...
MARC:
Solve the eqn. \(x^{2}+6x-3=0\)
MARC:
Use the quadratic formula to solve that eqn.
MARC:
\(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\)
MARC:
\(\color{red}{a}x^{2}+\color{blue}{b}x+\color{green}{c}=0\)
MARC:
\(\color{red}{1}x^{2}+(\color{blue}{-6}x)+(\color{green}{-3})=0\)
MARC:
a=1 b=-6 c=-3
MARC:
sub. these values into the quadractic formula.
MARC:
@MARC
MARC:
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MARC:
@MARC
MARC:
@MARC
MARC:
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MARC:
@MARC
MARC:
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