Question 12 Marks
$2-\sqrt{5},2+\sqrt{5}$
Answer$\text { Let } \alpha=2-\sqrt{5} \text { and } \beta=2+\sqrt{5}$
$\therefore \alpha+\beta=2-\sqrt{5}+2+\sqrt{5}=4 \text { and } \alpha \beta=(2-\sqrt{5})(2+\sqrt{5})=4-5=1$
$\therefore$ and quadratic equation is, $x^2-(\alpha+\beta) x+\alpha \beta=0$
$\therefore x^2-(4) x+(1)=0 $
$\therefore x^2-4 x+1=0$
View full question & answer→Question 22 Marks
$\frac{1}{2},-\frac{1}{2}$
AnswerLet $\alpha=\frac{1}{2}$ and $\beta=-\frac{1}{2}$
$\therefore \alpha+\beta=\frac{1}{2}-\frac{1}{2}=0 \text { and } \alpha \beta=\frac{1}{2} \times-\frac{1}{2}=-\frac{1}{4}$
$\therefore$ and quadratic equation is, $x ^2-(\alpha+\beta) x +\alpha \beta=0$
$\therefore x ^2-(0) x +\left(-\frac{1}{4}\right)=0$
$\therefore x ^2-\frac{1}{4}=0 $
$\therefore 4 x ^2-1=0$
View full question & answer→Question 32 Marks
AnswerLet $\alpha=3$ and $\beta=-10$
$\therefore \alpha+\beta=3-10=-7 \text { and } \alpha \beta=3 \times-10=-30$
$\therefore$ and quadratic equation is, $x ^2-(\alpha+\beta) x +\alpha \beta=0$
$\therefore x ^2-(-7) x +(-30)=0$
$\therefore x ^2+7 x -30=0$
View full question & answer→Question 42 Marks
AnswerLet $\alpha=0$ and $\beta=4$
$\therefore \alpha+\beta=0+4=4 \text { and } \alpha \beta=0 \times 4=0$
$\therefore$ and quadratic equation is, $x^2-(\alpha+\beta) x+\alpha \beta=0$
$\therefore x ^2-(4) x +(0)=0 $
$\therefore x ^2-4 x =0$
View full question & answer→Question 52 Marks
Determine the nature of roots of the following quadratic equation.
$m^2+2 m+9=0$
Answer$m ^2+2 m+9=0$ compare with $ax ^2+ bx + c =0$
$\Rightarrow a =1, b=2$ and $c =9$
$\therefore b ^2-4 ac =2^2-4(1)(9)$
$=4-36$
$=-32$
$\therefore b ^2-4 ac <0$.hence, roots are not real.
View full question & answer→Question 62 Marks
Determine the nature of roots of the following quadratic equation.
$2 y^2-7 y+2=0$
Answer$2 y ^2-7 y +2=0 \text { compare with } ax ^2+ bx + c =0 $
$ \Rightarrow a =2, b=-7 \text { and } c =2 $
$ \therefore b ^2-4 ac =-7^2-4(2)(2) $
$=49-16 $
$ =23$
$ \therefore b ^2-4 ac >0 . \text { Hence, roots are real and unequal }$
View full question & answer→Question 72 Marks
Determine the nature of roots of the following quadratic equation.
$x^2-4 x+4=0$
Answer$x ^2-4 x +4=0 \text { compare with } ax ^2+ bx + c =0$
$ \Rightarrow a =1, b=-4 \text { and } c =4 $
$\therefore b ^2-4 ac =-4^2-4(1)(4) $
$ =16-16 $
$ =0 $
$ \therefore b ^2-4 ac =0 \text {.hence,roots are real and equal }$
View full question & answer→Question 82 Marks
Find the value of discriminant.
$\sqrt{2} x ^2+4 x +2 \sqrt{2}=0$
Answer$\sqrt{2} x ^2+4 x +2 \sqrt{2}=0 \text { compare with } ax ^2+ bx + c =0 $
$ \Rightarrow a =\sqrt{2}, b=4 \text { and } c =2 \sqrt{2} $
$\therefore b ^2-4 ac =4^2-4(\sqrt{2})(2 \sqrt{2}) $
$ =16-16 $
$ =0$
View full question & answer→Question 92 Marks
Find the value of discriminant.
$2y^2 – 5y + 10 = 0$
View full question & answer→Question 102 Marks
Find the value of discriminant.
$x^2+7 x-1=0$
Answer$x^2+7 x-1=0 \text { compare with } ax^2+b x+c=0$
$ \Rightarrow a=1, b=7 \text { and } c=-1 $
$\therefore b^2-4 a c=7^2-4(1)(-1) $
$ =49+4$
$ =53$
View full question & answer→Question 112 Marks
Solve the following quadratic equation by factorization.
2m (m-24) = 50
Answer$2 m(m-24)=50 $
$2 m^2-48 m-50=0$
$\Rightarrow 2 m^2-50 m+2 m-50=0 $
$\Rightarrow 2 m(m-25)+2(m-25)=0 $
$\Rightarrow(2 m+2)(m-25)=0$
$\Rightarrow 2 m+2=0 \text { or } m-25=0 $
$\Rightarrow m=-1 \text { or } m=25$
Hence, $m =-1$ or $m =25$ are roots of the equation.
View full question & answer→Question 122 Marks
Solve the following quadratic equation by factorization.
$3 x^2-2 \sqrt{6} x+2=0$
Answer$3 x^2-\sqrt{6} x-\sqrt{6} x+2=0 $
$ \Rightarrow \sqrt{3} x(\sqrt{3} x-\sqrt{2})-\sqrt{2}(\sqrt{3} x-\sqrt{2})=0$
$ \Rightarrow(\sqrt{3} x-\sqrt{2})(\sqrt{3} x-\sqrt{2})=0$
$\Rightarrow(\sqrt{3} x-\sqrt{2})=0 \text { or }(\sqrt{3} x-\sqrt{2})=0$
$ x=\frac{\sqrt{2}}{\sqrt{3}} \text { or } x=\frac{\sqrt{2}}{\sqrt{3}}$
View full question & answer→Question 132 Marks
Solve the following quadratic equation by factorization.
$\sqrt{2 x^2}+7 x+5 \sqrt{2}=0 $ to solve this quadratic equation by factorization, complete the following activity.
Answer$\sqrt{2 x^2}+7 x+5 \sqrt{2}=0 $
$\sqrt{2} x^2+5 x+2 x+5 \sqrt{2}=0 $
$x(\sqrt{2} x+5)+\sqrt{2}(\sqrt{2} x+5)=0 $
$(x+\sqrt{2})(\sqrt{2} x+5)=0 $
$(x+\sqrt{2})=0 \text { or }(\sqrt{2} x+5)=0 $
$x=-\frac{5}{\sqrt{2}} \text { or } x=-\sqrt{2}$
$\therefore-\frac{5}{\sqrt{2}}$ and $-\sqrt{2}$ are roots of the equation.
View full question & answer→Question 142 Marks
Solve the following quadratic equation by factorization.
$6x - \frac{2}{x} = 1$
Answer$6 x^2-2=x $
$\Rightarrow 6 x^2-x-2=0 $
$\Rightarrow 6 x^2+3 x-4 x-2=0 $
$\Rightarrow 3 x(2 x+1)-2(2 x+1)=0 $
$\Rightarrow(3 x-2)(2 x+1)=0 $
$3 x-2=0 \Rightarrow 3 x=2 \Rightarrow x=\frac{2}{3} $
$2 x+1=0 \Rightarrow 2 x=-1 \Rightarrow x=-\frac{1}{2}$
Hence, $x =\frac{2}{3}$ and $x =-\frac{1}{2}$ are roots of the equation.
View full question & answer→Question 152 Marks
Solve the following quadratic equation by factorization.
$2 x ^2-2 x +\frac{1}{2}=0 $
Answer$2 x ^2-2 x +\frac{1}{2}=0 $
$\Rightarrow 4 x ^2-4 x +1=0 $
$\Rightarrow 4 x ^2-2 x -2 x +1 $
$\Rightarrow 2 x(2 x-1)-1(2 x-1) $
$\Rightarrow(2 x-1)(2 x-1) $
$\Rightarrow 2 x-1=0 \Rightarrow x=\frac{1}{2}, \frac{1}{2}$
Hence $x=\frac{1}{2}, \frac{1}{2}$ are roots of the equation
View full question & answer→Question 162 Marks
Solve the following quadratic equation by factorization.
$5 m^2=22 m+15$
Answer$5 m^2-22 m-15=0 $
$\Rightarrow 5 m^2-3 m+25 m-15 $
$\Rightarrow m (5 m-3)+5(5 m-3) $
$\Rightarrow( m +5)(5 m-3) $
$m +5=0 \Rightarrow m =-5$
$5 m-3=0 \Rightarrow 5 m=3 \Rightarrow m =\frac{3}{5}$
$\therefore$ Hence, $m =-5$ and $m =\frac{3}{5}$ are roots of the equation.
View full question & answer→Question 172 Marks
Solve the following quadratic equation by factorization.
$2y^2 + 27y + 13 = 0$
Answer$2 y^2+27 y+13=0 $
$\Rightarrow 2 y^2+26 y+y+13=0 $
$\Rightarrow 2 y(y+13)+(y+13)=0 $
$\Rightarrow(2 y+1)(y+13)=0 $
$2 y+1=0 \Rightarrow 2 y=-1 \Rightarrow y=-\frac{1}{2} $
$y+13=0 \Rightarrow y=-13$
Hence, $y =-13$ and $y =-\frac{1}{2}$ are roots of the equation.
View full question & answer→Question 182 Marks
Solve the following quadratic equation by factorization.
$x^2+x-20=0$
Answer$x^2+x-20=0 $
$\Rightarrow x^2+5 x-4 x-20=0$
$\Rightarrow x(x+5)-4(x+5)=0$
$\Rightarrow(x+5)(x-4)=0 $
$x+5=0 \Rightarrow x=-5$
$x-4=0 \Rightarrow x=4$
Hence, $x =-5$ and $x =4$ are roots of the equation.
View full question & answer→Question 192 Marks
Solve the following quadratic equation by factorization.
$m^2-11=0$
Answer$m ^2-11=0 $
$\Rightarrow m ^2=11 $
$\Rightarrow m =\sqrt{11}$
$\Rightarrow m = \pm 11$
Hence,$m= \pm 11$ are roots of the equation.
View full question & answer→Question 202 Marks
Solve the following quadratic equation by factorization.
$7 m^2=21 m$
Answer$7 m^2-21 m=0$
$\Rightarrow 7 m( m -3)=0 $
$\Rightarrow 7 m=0 \text { or } m -3=0 $
$\Rightarrow m =0 \text { or } m =3$
Hence, $m =0$ or $m =3$ are roots of the equation.
View full question & answer→Question 212 Marks
Solve the following quadratic equation by factorization.
$25 m^2=9$
Answer$25 m^2=9 $
$\Rightarrow m ^2=\frac{9}{25}$
$\Rightarrow m =\sqrt{\frac{9}{25}} $
$\Rightarrow m = \pm \frac{3}{5}$
Hence, $m= \pm \frac{3}{5}$ are roots of the equation.
View full question & answer→Question 222 Marks
Solve the following quadratic equation by factorization.
$x^2-15 x+54=0$
Answer
$x^2-15 x+54=0 $
$\Rightarrow x^2-6 x-9 x+54=0$
$\Rightarrow x(x-6)-9(x-6)=0 $
$\Rightarrow(x-6)(x-9)=0 $
$x-6=0 \Rightarrow x=6 $
$x-9=0 \Rightarrow x=9$
Hence, $x =6$ and $x =9$ are roots of the equation.
View full question & answer→Question 232 Marks
Determine the nature of roots for each of the quadratic equation.
$m^2-2 m+1=0$
Answer$m ^2-2 m+1=0 \text { compare with } ax ^2+ bx + c =0 $
$ \Rightarrow a =1, b=-2 \text { and } c =1 $.
$ \therefore b ^2-4 ac =-2^2-4(1)(1) $
$ =4-4 $
$ =0 $
$ \therefore b ^2-4 ac =0 $.hence,roots are real and equal.
View full question & answer→Question 242 Marks
Determine the nature of roots for each of the quadratic equation.
$\sqrt{3} x^2+\sqrt{2} x-2 \sqrt{3}=0$
Answer$\sqrt{3} x^2+\sqrt{2} x+2 \sqrt{3}=0 \text { compare with } a x^2+b x+c=0 $
$\Rightarrow a=\sqrt{3}, b=\sqrt{2} \text { and } c=-2 \sqrt{3} $
$\therefore b^2-4 a c=\sqrt{2}^2-4(\sqrt{3})(-2 \sqrt{3}) $
$=2+24 $
$=26$
$\therefore b ^2-4 ac >0$.hence, roots are real and umequal.
View full question & answer→Question 252 Marks
Determine the nature of roots for each of the quadratic equation.
$3 x^2-5 x+7=0$
Answer$3 x^2-5 x+7=0$ compare with $a x^2+b x+c=0$
$\Rightarrow a =3, b=-5$ and $c =7$
$\therefore b ^2-4 ac =-5^2-4(3)(7)$
$=25-147$
$=-122$
$\therefore b ^2-4 ac <0$.hence, roots are not real.
View full question & answer→Question 262 Marks
Find k if $x =3$ is a root of equation $k x^2-10 x +3=0$.
Answer$ kx ^2-10 x +3=0 \text { Put } x =3 $
$ \Rightarrow k (3)^2-10 \times 3+3=0 $
$ \Rightarrow 9 k -30+3=0$
$\Rightarrow 9 k =30-3 $
$ \Rightarrow 9 k =27 $
$ \Rightarrow k =\frac{27}{9}=3$
View full question & answer→Question 272 Marks
One of the roots of quadratic equation $2 x^2+k x-2=0$ is -2 , find $k$.
Answer$2 x ^2+ kx -2=0 $
$ \Rightarrow 2 \times-2^2-2 k -2=0$
$ \Rightarrow 8-2-2 k =0$
$\Rightarrow 6=2 k$
$ k =3$
View full question & answer→Question 282 Marks
Find the value of discriminant for each of the following equation.
$\sqrt{5} x^2-x-\sqrt{5}=0$
Answer$\sqrt{5} x^2-x-\sqrt{5}=0 \text { compare with } a x^2+b x+c=0 $
$ \Rightarrow a=\sqrt{5}, b=-1 \text { and } c=-\sqrt{5} $
$ \therefore b^2-4 a c=-1^2-4(\sqrt{5})(-\sqrt{5}) $
$ =1+20 $
$ =21$
View full question & answer→Question 292 Marks
Find the value of discriminant for each of the following equation.
$5 m^2-m=0$
Answer$5 m^2- m =0 \text { compare with } ax ^2+ bx + c =0 $
$ \Rightarrow a =5, b=-1 \text { and } c =0 $
$ \therefore b ^2-4 ac =-1^2-4(5)(0) $
$ =1$
View full question & answer→Question 302 Marks
Find the value of discriminant for each of the following equation.
$2 y^2-y+2=0$
Answer$2 y^2-y+2=0 \text { compare with } ax ^2+ bx + c =0 $
$ \Rightarrow a =2, b=-1 \text { and } c =2 $
$ \therefore b ^2-4 ac =-1^2-4(2)(2) $
$ =1-16 $
$=-15
View full question & answer→Question 312 Marks
Obtain the quadratic equation if roots are $-3,-7$.
AnswerLet $\alpha=-3$ and $\beta=-7$
$ \therefore \alpha+\beta=(-3)+(-7)=-10 \text { and } \alpha \times \beta=(-3) \times(-7)=21$
$\therefore \text { and quadratic equation is, } x^2-(\alpha+\beta) x+\alpha \beta=0$
$\therefore x^2-(-10) x+21=0$
$\therefore x^2+10 x+21=0 $
View full question & answer→Question 322 Marks
Determine nature of roots of the quadratic equations : √3x² + 2√3x + 3 = 0
Question 332 Marks
Determine nature of roots of the quadratic equations : x² + 2x - 9 = 0
Question 342 Marks
Determine nature of roots of the quadratic equations : 2x² - 5x + 7 = 0
Question 352 Marks
View full question & answer→Question 362 Marks
View full question & answer→Question 372 Marks
$2 x^2-7 x+6=0$ check whether$x=-2$ are solutions of the equations.
AnswerLet $x=-2$ in $2 x^2-7 x+6$
\*
\begin{aligned}
2 x^2-7 x+6= & 2(-2)^2-7(-2)+6 \\
& =2 \times 4+14+6 \\
& =28 \neq 0
\end{aligned}
$
$\therefore x=-2$ is not a solution of the equation.
View full question & answer→Question 382 Marks
$2 x^2-7 x+6=0$ check whether (i) $x=\frac{3}{2}$,
AnswerPut $x=\frac{3}{2}$ in the polynomial $2 x^2-7 x+6$
$2 x^2-7 x+6=2\left(\frac{3}{2}\right)^2-7\left(\frac{3}{2}\right)+6$
$32$
$=2 \times \frac{9}{4}-\frac{21}{2}+6$
$=\frac{9}{2}-\frac{21}{2}+\frac{12}{2}=0 $
$\therefore x=\frac{3}{2}$ is a solution of the equation.
View full question & answer→Question 392 Marks
Solve the following quadratic equation by using formula method:
$9 y^2-5 y-4=0$
Answer1 and $\frac{-4}{9}$
View full question & answer→Question 402 Marks
Solve the following quadratic equation by using formula method:
$4 x^2+x-5=0$
Answer1 and $\frac{-5}{4}$
View full question & answer→Question 412 Marks
Solve the following quadratic equation by using formula method:
$x^2+4 x-1=0$
Answer$-2+\sqrt{5}$ and $-2-\sqrt{5}$
View full question & answer→Question 422 Marks
Solve the following quadratic equation by using formula method:
$5 m^2+5 m=1$
Answer$\frac{-5+3 \sqrt{5}}{10}$ and $\frac{-5-3 \sqrt{5}}{10}$
View full question & answer→Question 432 Marks
Solve the following quadratic equation by using formula method:
$3 x^2+8 x+3=0$
Answer$\frac{-4+\sqrt{7}}{3}$ and $\frac{-4-\sqrt{7}}{3}$
View full question & answer→Question 442 Marks
Solve the following quadratic equation by using formula method:
$x^2+2 x-7=0$
Answer$-1+2 \sqrt{2}$ and $-1-2 \sqrt{2}$
View full question & answer→Question 452 Marks
Solve the following quadratic equation by factorization method:
$m^2-7=0$
Answer$x=\sqrt{7}$ or $x=-\sqrt{7}$
View full question & answer→Question 462 Marks
Solve the following quadratic equation by factorization method:
$7 x^2+4 x-20=0$
Answer$x=-2$ or $x=\frac{10}{7}$
View full question & answer→Question 472 Marks
Solve the following quadratic equation by factorization method:
$x^2-3 \sqrt{3} x+6=0$
Answer$x=\sqrt{3}$ or $x=2 \sqrt{3}$
View full question & answer→Question 482 Marks
Solve the following quadratic equation by factorization method:
$16 x^2-24 x=0$
Answer$x=0$ or $x=\frac{3}{2}$
View full question & answer→Question 492 Marks
Solve the following quadratic equation by factorization method:
$x^2-13 x+30=0$
View full question & answer→Question 502 Marks
Solve the following quadratic equation by factorization method:
$64 m^2-625=0$
Answer$m=\frac{-25}{8}$ or $m=\frac{25}{8}$
View full question & answer→