Question
Solve the following equation by factorisation :
$
\sqrt{3 x^2-2 x-1}=2 x-2
$
$
\sqrt{3 x^2-2 x-1}=2 x-2
$
✓
Answer
$
\sqrt{3 x^2-2 x-1}=2 x-2
$
Squaring both sides
$
\begin{aligned}
& 3 x^2-2 x-1=(2 x-2)^2 \\
& \Rightarrow 3 x^2-2 x-1=4 x^2-8 x+4 \\
& \Rightarrow 4 x^2-8 x+4-3 x^2+2 x+1=0 \\
& \Rightarrow x^2-6 x+5=0 \\
& \Rightarrow x^2-5 x-x+5=0 \\
& \Rightarrow x(x-5)-1(x-5)=0 \\
& \Rightarrow(x-5)(x-1)=0
\end{aligned}
$
Either $x-5=0$,
$
\text { then } x=5
$
or
$
x-1=0
$
then $x=1$
Check:
$
\begin{aligned}
& \text { (i) If } x=5 \text {, then } \\
& \text { L.H.S. }=\sqrt{3 x^2-2 x-1} \\
& =\sqrt{3 \times(5)^2-2 \times 5-1} \\
& =\sqrt{3 \times 25-10-1} \\
& =\sqrt{75-10-1} \\
& =\sqrt{64} \\
& =8 \\
& \text { R.H.S. }=2 x-2 \\
& =2 \times 5-2 \\
& =10-2 \\
& =8 \\
& \because \text { L.H.S. }=\text { R.H.S. }
\end{aligned}
$
$\therefore x=5$ is a root
$
\begin{aligned}
& \text { (ii) If } x =1 \text {, then } \\
& \text { L.H.S. }=\sqrt{3 x^2-2 x-1} \\
& =\sqrt{3(1)^2-2(1)-1} \\
& =\sqrt{3 \times 1-2-1} \\
& =\sqrt{3-2-1} \\
& =0 \\
& \text { R.H.S. }=2 x-2 \\
& =2 \times 1-2 \\
& =2-2 \\
& =0 \\
& \because \text { L.H.S. }=\text { R.H.S. }
\end{aligned}
$
$\therefore x =1$ is also its root
Hence $x=5,1$.
\sqrt{3 x^2-2 x-1}=2 x-2
$
Squaring both sides
$
\begin{aligned}
& 3 x^2-2 x-1=(2 x-2)^2 \\
& \Rightarrow 3 x^2-2 x-1=4 x^2-8 x+4 \\
& \Rightarrow 4 x^2-8 x+4-3 x^2+2 x+1=0 \\
& \Rightarrow x^2-6 x+5=0 \\
& \Rightarrow x^2-5 x-x+5=0 \\
& \Rightarrow x(x-5)-1(x-5)=0 \\
& \Rightarrow(x-5)(x-1)=0
\end{aligned}
$
Either $x-5=0$,
$
\text { then } x=5
$
or
$
x-1=0
$
then $x=1$
Check:
$
\begin{aligned}
& \text { (i) If } x=5 \text {, then } \\
& \text { L.H.S. }=\sqrt{3 x^2-2 x-1} \\
& =\sqrt{3 \times(5)^2-2 \times 5-1} \\
& =\sqrt{3 \times 25-10-1} \\
& =\sqrt{75-10-1} \\
& =\sqrt{64} \\
& =8 \\
& \text { R.H.S. }=2 x-2 \\
& =2 \times 5-2 \\
& =10-2 \\
& =8 \\
& \because \text { L.H.S. }=\text { R.H.S. }
\end{aligned}
$
$\therefore x=5$ is a root
$
\begin{aligned}
& \text { (ii) If } x =1 \text {, then } \\
& \text { L.H.S. }=\sqrt{3 x^2-2 x-1} \\
& =\sqrt{3(1)^2-2(1)-1} \\
& =\sqrt{3 \times 1-2-1} \\
& =\sqrt{3-2-1} \\
& =0 \\
& \text { R.H.S. }=2 x-2 \\
& =2 \times 1-2 \\
& =2-2 \\
& =0 \\
& \because \text { L.H.S. }=\text { R.H.S. }
\end{aligned}
$
$\therefore x =1$ is also its root
Hence $x=5,1$.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
If I is the incentre of triangle ABC and AI when produced meets the cicrumcircle of triangle ABC in points D. f ∠BAC = 66° and ∠ABC = 80°. Calculate : ∠BIC.
A buoy is made in the form of hemisphere surmounted by a right cone whose circular basecoincides with the plane surface of hemisphere. The radius of the base of the cone is $3.5$ metres and its volume is two third of the hemisphere. Calculate the height of the cone and the surfacearea of the buoy, correct to two places of decimal
→The difference between the outer curved surface area and the inner curved surface area of a hollow cylinder is $352 cm^2$. If its height is $28 \ cm$ and the volume of material in it is $704 cm^3$; find its external curved surface area.
→→
A two digit number is such that its product of its digit is 18. When 63 is subtracted from the number, the digits interchange their places. Find the number.
An A.P consists of $57$ terms of which $7$th term is $13$ and the last term is $108.$ Find the $45$th term of this A.P.
→A company, with 10,000 shares of Rs 100 each, declares an annual dividend of 5%.
(1) What is the total amount of dividend paid by the company?
(2) What should be the annual income of a man who has 72 shares in the company?
(3) If he received only 4% of his investment, find the price he paid for each share
(1) What is the total amount of dividend paid by the company?
(2) What should be the annual income of a man who has 72 shares in the company?
(3) If he received only 4% of his investment, find the price he paid for each share
If $(2 x+1)$ is a factor of both the expressions $2 x^2-5 x+p$ and $2 x^2+5 x+q$, find the value of $p$ and $q$. Hence find the other factors of both the polynomials.
→In the given figure, from the top of a building $AB = 60\ m$ hight, the angle of depression of the top and bottom of a vertical lamp post $CD$ are observed to be $30^\circ $ and $60^\circ$ respectively. find :
$(1) $ the horizental distance between $AB $ and $CD$ ;
$(2)$ the height of the lamp post.
→$(1) $ the horizental distance between $AB $ and $CD$ ;
$(2)$ the height of the lamp post.
Find the values of a and b when the factors of the polynomial $f(x)= ax^3 + bx^2 + x a$ are $(x+3)$ and $(2x-1).$ Factorize the polynomial completely.
→