Find and correct the errors in the following.
(a) $ (2 x+5)^{2}=4 x^{2}+25 $
(b) $ \left(x-\frac{1}{2}\right)\left(x-\frac{1}{2}\right)=x^{2}-\frac{1}{4} $
(c) $ (5 a-b)^{2}=10 a^{2}-5 a b+b^{2} $
(d) $ (p-3)(p-7)=p^{2}+21 $
To do:
We have to correct the errors in the below expressions.
(a) \( (2 x+5)^{2}=4 x^{2}+25 \)
(b) \( \left(x-\frac{1}{2}\right)\left(x-\frac{1}{2}\right)=x^{2}-\frac{1}{4} \)
(c) \( (5 a-b)^{2}=10 a^{2}-5 a b+b^{2} \)
(d) \( (p-3)(p-7)=p^{2}+21 \)
Solution:
We know that,
$(a+b)^2=a^2+2ab+b^2$
$(a-b)^2=a^2-2ab+b^2$
Therefore,
(a) LHS $=(2x+5)^2$
$=(2x)^2+2.(2x).(5)+(5)^2$
$=4x^2+20x+25$
RHS $=4x^2+25$
Therefore, we have to add $20x$ on RHS
(b) LHS $=(x-\frac{1}{2})(x-\frac{1}{2})$
$=(x-\frac{1}{x})^2$
$=x^2-2.x.\frac{1}{x}+(\frac{1}{x})^2$
$=x^2-2+\frac{1}{x^2}$
RHS $=x^{2}-\frac{1}{4}$
Therefore, we have to replace $-\frac{1}{4}$ by $-2+\frac{1}{x^2}$ on RHS
(c) LHS $=(5 a-b)^{2}$
$=(5a)^2-2.5a.b+(b)^2$
$=25a^2-10ab+b^2$
RHS $=10 a^{2}-5 a b+b^{2}$
Therefore, we have to replace $10 a^{2}-5 a b$ by $25a^2-10ab$ on RHS.
(d) LHS $=(p-3)(p-7)$
$=p(p)-p(7)-3(p)-3(-7)$
$=p^2-7p-3p+21$
$=p^2-10p+21$
RHS $=p^{2}+21$
Therefore, we have to add $-10p$ on RHS.
Related Articles
- Find the product of the following binomials:(i) \( (2 x+y)(2 x+y) \)(ii) \( (a+2 b)(a-2 b) \)(iii) \( \left(a^{2}+b c\right)\left(a^{2}-b c\right) \)(iv) \( \left(\frac{4 x}{5}-\frac{3 y}{4}\right)\left(\frac{4 x}{5}+\frac{3 y}{4}\right) \)(v) \( \left(2 x+\frac{3}{y}\right)\left(2 x-\frac{3}{y}\right) \)(vi) \( \left(2 a^{3}+b^{3}\right)\left(2 a^{3}-b^{3}\right) \)(vii) \( \left(x^{4}+\frac{2}{x^{2}}\right)\left(x^{4}-\frac{2}{x^{2}}\right) \)(viii) \( \left(x^{3}+\frac{1}{x^{3}}\right)\left(x^{3}-\frac{1}{x^{3}}\right) \).
- (i) \( x^{2}-3 x+5-\frac{1}{2}\left(3 x^{2}-5 x+7\right) \)(ii) \( [5-3 x+2 y-(2 x-y)]-(3 x-7 y+9) \)(iii) \( \frac{11}{2} x^{2} y-\frac{9}{4} x y^{2}+\frac{1}{4} x y-\frac{1}{14} y^{2} x+\frac{1}{15} y x^{2}+ \) \( \frac{1}{2} x y \)(iv) \( \left(\frac{1}{3} y^{2}-\frac{4}{7} y+11\right)-\left(\frac{1}{7} y-3+2 y^{2}\right)- \) \( \left(\frac{2}{7} y-\frac{2}{3} y^{2}+2\right) \)(v) \( -\frac{1}{2} a^{2} b^{2} c+\frac{1}{3} a b^{2} c-\frac{1}{4} a b c^{2}-\frac{1}{5} c b^{2} a^{2}+ \) \( \frac{1}{6} c b^{2} a+\frac{1}{7} c^{2} a b+\frac{1}{8} c a^{2} b \).
- Subtract the following:\( \left(4 x^{2}-\frac{1}{5} x+7\right)-\left(-2 x^{2}-\frac{1}{2} x+\frac{1}{3}\right) \)
- Write the following squares of bionomials as trinomials:(i)\( (x+2)^{2} \)(ii) \( (8 a+3 b)^{2} \)(iii) \( (2 m+1)^{2} \)(iv) \( \left(9 a+\frac{1}{6}\right)^{2} \)(v) \( \left(x+\frac{x^{2}}{2}\right)^{2} \)(vi) \( \left(\frac{x}{4}-\frac{y}{3}\right)^{2} \)(vii) \( \left(3 x-\frac{1}{3 x}\right)^{2} \)(viii) \( \left(\frac{x}{y}-\frac{y}{x}\right)^{2} \)(ix) \( \left(\frac{3 a}{2}-\frac{5 b}{4}\right)^{2} \)(x) \( \left(a^{2} b-b c^{2}\right)^{2} \)(xi) \( \left(\frac{2 a}{3 b}+\frac{2 b}{3 a}\right)^{2} \)(xii) \( \left(x^{2}-a y\right)^{2} \)
- Simplify each of the following.(a) \( \left(\frac{1}{2} a-b\right)\left(\frac{1}{2} a+b\right)\left(\frac{1}{4} a^{2}+b^{2}\right) \)(b) \( \left(\frac{p}{2}-\frac{q}{3}\right)\left(\frac{p}{2}+\frac{q}{3}\right)\left(\frac{p^{2}}{4}+\frac{q^{2}}{9}\right)\left(\frac{p^{4}}{16}+\frac{a^{4}}{81}\right) \)(c) \( \left(a^{2}+1-a\right)\left(a^{2}-1+a\right) \)(d) \( \left(4 b^{2}+2 b+1\right)\left(4 b^{2}-2 b-1\right) \)
- Find $x$, if(i) \( \left(\frac{1}{4}\right)^{-4} \times\left(\frac{1}{4}\right)^{-8}=\left(\frac{1}{4}\right)^{-4 x} \)(ii) \( \left(\frac{-1}{2}\right)^{-19}\div\left(\frac{-1}{2}\right)^{8}=\left(\frac{-1}{2}\right)^{-2 x+1} \)(iii) \( \left(\frac{3}{2}\right)^{-3} \times\left(\frac{3}{2}\right)^{5}=\left(\frac{3}{2}\right)^{2 x+1} \)(iv) \( \left(\frac{2}{5}\right)^{-3} \times\left(\frac{2}{5}\right)^{15}=\left(\frac{2}{5}\right)^{2+3 x} \)(v) \( \left(\frac{5}{4}\right)^{-x}\div\left(\frac{5}{4}\right)^{-4}=\left(\frac{5}{4}\right)^{5} \)(vi) \( \left(\frac{8}{3}\right)^{2 x+1} \times\left(\frac{8}{3}\right)^{5}=\left(\frac{8}{3}\right)^{x+2} \)
- Add the following algebraic expressions(i) \( 3 a^{2} b,-4 a^{2} b, 9 a^{2} b \)(ii) \( \frac{2}{3} a, \frac{3}{5} a,-\frac{6}{5} a \)(iii) \( 4 x y^{2}-7 x^{2} y, 12 x^{2} y-6 x y^{2},-3 x^{2} y+5 x y^{2} \)(iv) \( \frac{3}{2} a-\frac{5}{4} b+\frac{2}{5} c, \frac{2}{3} a-\frac{7}{2} b+\frac{7}{2} c, \frac{5}{3} a+ \) \( \frac{5}{2} b-\frac{5}{4} c \)(v) \( \frac{11}{2} x y+\frac{12}{5} y+\frac{13}{7} x,-\frac{11}{2} y-\frac{12}{5} x-\frac{13}{7} x y \)(vi) \( \frac{7}{2} x^{3}-\frac{1}{2} x^{2}+\frac{5}{3}, \frac{3}{2} x^{3}+\frac{7}{4} x^{2}-x+\frac{1}{3} \) \( \frac{3}{2} x^{2}-\frac{5}{2} x-2 \)
- Show that:\( \left(\frac{x^{a^{2}+b^{2}}}{x^{a b}}\right)^{a+b}\left(\frac{x^{b^{2}+c^{2}}}{x^{b c}}\right)^{b+c}\left(\frac{x^{c^{2}+a^{2}}}{x^{a c}}\right)^{a+c}= x^{2\left(a^{3}+b^{3}+c^{3}\right)} \)
- Prove that:\( \left(\frac{x^{a}}{x^{b}}\right)^{a^{2}+a b+b^{2}} \times\left(\frac{x^{b}}{x^{c}}\right)^{b^{2}+b c+c^{2}} \times\left(\frac{x^{c}}{x^{a}}\right)^{c^{2}+c a+a^{2}}=1 \)
- Prove that:\( \left(\frac{x^{a}}{x^{-b}}\right)^{a^{2}-a b+b^{2}} \times\left(\frac{x^{b}}{x^{-c}}\right)^{b^{2}-b c+c^{2}} \times\left(\frac{x^{c}}{x^{-a}}\right)^{c^{2}-c a+a^{2}}=1 \)
- Solve the following:\( \frac{3}{4}(7 x-1)-\left(2 x-\frac{1-x}{2}\right)=x+\frac{3}{2} \)
- Take away:(i) \( \frac{6}{5} x^{2}-\frac{4}{5} x^{3}+\frac{5}{6}+\frac{3}{2} x \) from \( \frac{x^{3}}{3}-\frac{5}{2} x^{2}+ \) \( \frac{3}{5} x+\frac{1}{4} \)(ii) \( \frac{5 a^{2}}{2}+\frac{3 a^{3}}{2}+\frac{a}{3}-\frac{6}{5} \) from \( \frac{1}{3} a^{3}-\frac{3}{4} a^{2}- \) \( \frac{5}{2} \)(iii) \( \frac{7}{4} x^{3}+\frac{3}{5} x^{2}+\frac{1}{2} x+\frac{9}{2} \) from \( \frac{7}{2}-\frac{x}{3}- \) \( \frac{x^{2}}{5} \)(iv) \( \frac{y^{3}}{3}+\frac{7}{3} y^{2}+\frac{1}{2} y+\frac{1}{2} \) from \( \frac{1}{3}-\frac{5}{3} y^{2} \)(v) \( \frac{2}{3} a c-\frac{5}{7} a b+\frac{2}{3} b c \) from \( \frac{3}{2} a b-\frac{7}{4} a c- \) \( \frac{5}{6} b c \)
- Multiply:\( \left(3 x^{2} y-5 x y^{2}\right) \) by \( \left(\frac{1}{5} x^{2}+\frac{1}{3} y^{2}\right) \)
- Choose the correct answer from the given four options in the following questions:Which of the following is a quadratic equation?(A) \( x^{2}+2 x+1=(4-x)^{2}+3 \)(B) \( -2 x^{2}=(5-x)\left(2 x-\frac{2}{5}\right) \)(C) \( (k+1) x^{2}+\frac{3}{2} x=7 \), where \( k=-1 \)(D) \( x^{3}-x^{2}=(x-1)^{3} \)
- (i) If \( x=\left(\frac{3}{2}\right)^{2} \times\left(\frac{2}{3}\right)^{-4} \), find the value of \( x^{-2} \)(ii) If \( x=\left(\frac{4}{5}\right)^{-2}\div\left(\frac{1}{4}\right)^{2} \), find the value of \( x^{-1} \).
Kickstart Your Career
Get certified by completing the course
Get Started
Data Structure
Networking
RDBMS
Operating System
Java
MS Excel
iOS
HTML
CSS
Android
Python
C Programming
C++
C#
MongoDB
MySQL
Javascript
PHP
Physics
Chemistry
Biology
Mathematics
English
Economics
Psychology
Social Studies
Fashion Studies
Legal Studies