Algebra: Quadratic Equations Test-2

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Algebra: Quadratic Equations Test-2

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Question 1

The minimum value of (x -2) (x-9) is

A	-11/4
B	49/4
C	0
D	-49/4

Question 1 Explanation:

$\displaystyle \begin{array}{l}\left( x\text{ }-2 \right)\text{ }\left( x-9 \right)\\={{x}^{2}}-11x+18\\={{x}^{2}}-2.\frac{11}{2}x+\frac{121}{4}-\frac{49}{4}\\={{(x-\frac{11}{2})}^{2}}-\frac{49}{4}\end{array}$
The minimum value of the square is 0. The minimum value is -49/4

Question 2

One of the factors of the expression

$\displaystyle 4\sqrt{3}{{x}^{2}}+5x-2\sqrt{3}$

A	$\displaystyle 4x+\sqrt{3}$
B	$\displaystyle 4x+3$
C	$\displaystyle 4x-3$
D	$\displaystyle 4x-\sqrt{3}$

Question 2 Explanation:

$\displaystyle \begin{array}{l}4\sqrt{3}\,\,\,{{x}^{2}}+5x-2\sqrt{3}\\=4\sqrt{3}\,\,\,{{x}^{2}}+8x-3x-2\sqrt{3}\\=4x\left( \sqrt{3}\,\,\,x+2 \right)-\sqrt{3}\left( \sqrt{3}x+2 \right)\\=\left( 4x-\sqrt{3} \right)\left( \sqrt{3}x+2 \right)\end{array}$

Question 3

$\displaystyle \begin{array}{l}\sqrt{x}=\sqrt{3}-\sqrt{5},\,\,\,\\then\,\,\,the\,\,\,value\,\,\,of\,\,\,\\{{x}^{2}}-16x+6\,\,is\end{array}$

A	0
B	-2
C	2
D	4

Question 3 Explanation:

$\sqrt{x}=\sqrt{3}-\sqrt{5}$

$\begin{array}{l}x=3+5-2\sqrt{15}\\=>x=8-2\sqrt{15}\\=>{{x}^{2}}=64+60-32\sqrt{15}=124-32\sqrt{15}\\Thus\,{{x}^{2}}-16x+6\\=124-32\sqrt{15}-16(8-2\sqrt{15})+6\\=124-32\sqrt{15}-128+32\sqrt{15}+6\\=2\end{array}$

Question 4

$\displaystyle x=\sqrt[3]{5}+2,\,\,then\,\,\,the\,\,value\,\,\,of\,\,\,{{x}^{3}}-6{{x}^{2}}+12x-13\,\,is$

A	-1
B	1
C	4
D	0

Question 4 Explanation:

$\begin{array}{l}\,{{x}^{3}}-6{{x}^{2}}+12x-13\\=\,{{x}^{3}}-3.2.{{x}^{2}}+{{3.2}^{2}}x-{{2}^{3}}-5\\={{(x-2)}^{3}}-5\\={{(\sqrt[3]{5}+2-2)}^{3}}-5\\=5-5\\=0\end{array}$

Question 5

A boy was asked of his age by his friend. The boy said, "The number you get when you subtract 25 times my age from twice the square of my age will be thrice your age." If the friend's age is 14, then the age of the boy is?

A	28 yr
B	21 yr
C	14 yr
D	25 yr

Question 5 Explanation:

$\begin{array}{l}Let\,the\,age\,be\,x.\\2{{x}^{2}}-25x=3X14\\=>2{{x}^{2}}-25x-42=0\\=>(x-14)(2x-3)=0\\=>x=14\,\end{array}$
Rejecting the negative solution, we find that the age is 14 years.

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