[2 Mark Question Answer]

Question 12 Marks

Determine the nature of the roots of the following quadratic equation :
$x^2 -5x+ 7= 0$

Answer

$x^2 -5x+ 7= 0$
$b^2 - 4ac$
$= (-5)^2 - 4(1)(7)$
$= 25 - 28$
$= -3$
Since discriminant is negative, hence the roots are imaginary.

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Question 22 Marks

Determine the nature of the roots of the following quadratic equation :
$(x - 1)(2x - 7) = 0$

Answer

$(x - 1)(2x - 7) = 0$
$2x^2 - 2x - 7x + 7 = o$
$2x^2 - 9x + 7 = o$
$b^2 - 4ac$
$= (-9)^2 - 4(2)(7)$
$= 81 - 56$
$= 25$
Since discriminant is a perfect square $1$ hence the roots are real and rational.

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Question 32 Marks

Determine the nature of the roots of the following quadratic equation :
$2x^2 -3x+ 4= 0$

Answer

$2x^2 -3x+ 4= 0$
$b^2 - 4ac$
$= (-3)^2 - 4(2)( 4)$
$= 9 - 32$
$= - 23$
Since discriminant is negative, hence the roots are imaginary.

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Question 42 Marks

Determine the nature of the roots of the following quadratic equation :
$2x^2 + 5x - 6 = 0$

Answer

$2x^2 + 5x - 6 = 0$
$b^2 - 4ac$
$= (5)^2 - 4(2)(-6)$
$= 25 + 48$
$= 73$
Since discriminant is positive, hence the roots are real and irrational.

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Question 52 Marks

Determine the nature of the roots of the following quadratic equation :
$4x^2 - 8x + 5 = 0$

Answer

$4x^2 - 8x + 5 = 0$
$b^2 - 4ac$
$= (-8)^2 - 4( 4)(5)$
$= 64 - 100$
$= - 36$
Since discriminant is negative, hence the roots are imaginary.

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Question 62 Marks

Determine the nature of the roots of the following quadratic equation :
$x^2 +3x+1=0$

Answer

$x^2 +3x+1=0$
$b^2 - 4ac$
$= (3)^2 - 4(1)(1)$
$= 9 - 4$
$= 5$
Since discriminant is positive, hence the roots are real and irrational.

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Question 72 Marks

Determine the nature of the roots of the following quadratic equation
$x^2 -4x + 4=0$

Answer

$x^2 -4x + 4=0$
$b^2 - 4ac$
$= (-4)^2- 4(1)(4)$
$= 16- 16$
$=0$
Since discriminant is $01$ hence the roots are real and equal .

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Question 82 Marks

Determine the nature of the roots of the following quadratic equation :
$2x^2 + x-1=0$

Answer

$2x^2 + x-1=0$
$b^2 - 4ac$
$= (1)^2-4(2)(-1)$
$= 1+ 8$
$=9$
Since $9$ is a perfect square and greater than $0$, hence the roots are real and rational.

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Question 92 Marks

Find the value of the discriminant in the following quadratic equation :
$4 \sqrt{3} x ^2+5 x -2 \sqrt{3}=0$

Answer

$4 \sqrt{3} x ^2+5 x -2 \sqrt{3}=0 $
$ \text { Discriminant }= b ^2-4 ac $
$ =(5)^2-4(4 \sqrt{3})(-2 \sqrt{3})$
$ =25+96 $
$ =121$

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Question 102 Marks

Find the value of the discriminant in the following quadratic equation:
$x^2 +2x-2=0$

Answer

$x^2 +2x-2=0$
Discriminant $= b^2 - 4ac$
$= (2)^2 - 4(1)(-2)$
$= 4 + 8$
$= 12$

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Question 112 Marks

Find the value of the discriminant in the following quadratic equation :
$10 x -\frac{1}{x}=3$

Answer

$10 x -\frac{1}{x}=3$
$10 x ^2-3 x -1=0 $
$\text { Discriminant }= b ^2-4 ac$
$ =(-3)^2-4(10)(-1) $
$=9+40 $
$ =49$

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Question 122 Marks

Find the value of the discriminant in the following quadratic equation:
$2x^2 - 3x + 1 = O$

Answer

$2x^2 - 3x + 1 = O$
Discriminant $= b^2 - 4ac$
$= (-3)^2 - 4(2)(1)$
$= 9 - 8$
$= 1$

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Question 132 Marks

Find the value of the discriminant in the following quadratic equation :
$x^2 +2x+4=0$

Answer

$x^2 +2x+4=0$
Discriminant $= b^2 - 4ac$
$= (2)^2 - 4(1)(4)$
$= 4 -16$
$= - 12$

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Question 142 Marks

Find the value of the discriminant in the following quadratic equation:
$2x^2 - 5x + 3 = 0$

Answer

$2x^2 - 5x + 3 = 0$
Discriminant- $b^2 - 4ac$
$(-5)^2 - 4(2)(3)$
$= 25 -24$
$= 1$

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Question 152 Marks

Solve the following equation $:a^2 b^2 x^2+b^2 x-a^2 x-1=0 $

Answer

$a^2 b^2 x^2+b^2 x-a^2 x-1=0 $
$b^2 x\left(a^2 x+1\right)-1\left(a^2 x+1\right)=0 $
$ \left(a^2 x+1\right)\left(b^2 x-1\right)=0 $
$ x=-\frac{1}{a^2}, x=\frac{1}{b^2}$

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Question 162 Marks

Solve the following equation: $abx^2 +(b^2-ac) x - bc = 0$

Answer

$a b x^2+\left(b^2-a c\right) x-b c=0$
$a b x^2+b^2 x-a c x-b c=0 $
$ b x(a x+b)-c(a x+b)=0 $
$ (a x+b)(b x-c)=0 $
$ x=-\frac{b}{a}, x=\frac{c}{b}$

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Question 172 Marks

Solve the following equation: $x^2+\left(a+\frac{1}{a}\right) x+1=0$

Answer

$x^2+\left(a+\frac{1}{a}\right) x+1=0 $
$x^2+a x+\frac{1}{a} x+1=0 $
$ x(x+a)+\frac{1}{a}(x+a)=0$
$ (x+a)\left(x+\frac{1}{a}\right)=0 $
$ x=-a, x=-\frac{1}{a}$

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Question 182 Marks

Solve the Following Equation : $x^2- x - a (a + 1) = o$

Answer

$x^2- x - a (a + 1) = o$
$x^2 + ax - (a + 1) x - c ( a + 1) = 0$
$x (x + a) - (a + 1) {(x + a)} = 0$
$(x+ a) {x - (a +1)} = 0$
$x = -a , x = (a +1)$

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Question 192 Marks

Solve the following equation :$ x^2 + 2ab = (2a + b)x$

Answer

$x^2 + 2ab = (2a + b )x$
$x^2 + 2ab = 2ax +bx$
$x^2 - 2ax - bx+ 2ab = 0$
$x (x - 2a) - b(x - 2a)= 0$
$(x - 2a)(x - b)= 0$
$x = 2a, x = b$

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Question 202 Marks

Solve the following equation: $4x^2 + 16x = 0$

Answer

$4x^2 + 16x = 0$
$4x(x + 4) =0$
$4x = 0, (x + 4) =0$
$x = 0, x = - 4$

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Question 212 Marks

Solve the following equation: $\frac{m}{n} x^2+\frac{n}{m}=1-2 x$

Answer

$\frac{m}{n} x^2+\frac{n}{m}=1-2 x$
$ \text { Multiply by } m n $
$ m^2 x^2+n^2=m n-2 m n x $
$ (m x+n)^2=m n $
$m x+n= \pm \sqrt{m n} $
$ m x=-n \pm \sqrt{m n} $
$ x=\frac{-n \pm \sqrt{m n}}{m}$

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Question 222 Marks

Solve the following equation:
$(2x+3) (3x-7) = 0$

Answer

$(2 x+3)(3 x-7)=0$
$ \Rightarrow(2 x+3)=(3 x-7)=0$
$ \Rightarrow 2 x=-3,3 x=7$
$\Rightarrow x=-\frac{3}{2}, x=\frac{7}{3}$

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