(i) Given equation is 2x2−3x−5=0
On comparing with ax2+bx+c=0, we get a=2,b=−3 and c=−5
Quadratic formula, x=−b±√b2−4ac2a,
=−(−3)±√(−3)2−4(2)(−5)2(2)
=3±√9+404
=3±√494=3±74
⇒x=3+74 or x=3−74
⇒x=104 or x=−44
⇒x=52 or x=−1
Hence, the roots of the given equation are 52 and −1.
(ii) Given equation is 5x2+13x+8=0
On comparing with ax2+bx+c=0, we get a=5,b=13 and c=8
Quadratic formula, x=−b±√b2−4ac2a,
=−(13)±√(13)2−4(5)(8)2(5)
=−13±√169−16010
=−13±√910=−13±310
⇒x=−13+310 or x=−13−310
⇒x=−1010 or x=−1610
⇒x=−1 or x=−85
Hence, the roots of the given equation are −1 and −85.
(iii) Given equation is −3x2+5x+12=0
On comparing with ax2+bx+c=0, we get a=−3,b=5 and c=12
Quadratic formula, x=−b±√b2−4ac2a,
=−5±√(5)2−4(−3)(12)2(−3)
=−5±√25+144−6=−5±√169−6
=−5±13−6
⇒x=(−5+13−6) or x=(−5−13−6)
⇒x=8−6 or x=−18−6
⇒x=−43 or x=3
Hence, the roots of the given equation are −43 and 3.
(iv) Given equation is −x2+7x−10=0
On comparing with ax2+bx+c=0, we get a=−1,b=7 and c=−10
Quadratic formula, x=−b±√b2−4ac2a,
=−7±√(7)2−4(−1)(−10)2(−1)
=−7±√49−40−2=−7±√9−2
=−7±3−2
⇒x=−7+3−2 or x=−7−3−2
⇒x=−4−2 or x=(−10)(−2)
⇒x=2 or x=5
Hence, the roots of the given equation are 2 and 5.
(v) Given equation is x2+2√2x−6=0
On comparing with ax2+bx+c=0, we get a=1,b=2√2 and c=−6
Quadratic formula, x=−b±√b2−4ac2a,
Therefore, roots of given quadratic equation is given by
x=−(2√2)±√(2√2)2−4(1)(−6)2(1)
=−2√2±√8+242
=−2√2±√16×22=−2√2±4√22
=−√2±2√2
⇒x=−√2+2√2 or x=−√2−2√2
⇒x=√2 or x=−3√2
Hence, the roots of the given equation are √2 and −3√2
(vi) Given equation is x3−3√5x+10=0
On comparing with ax2+bx+c=0, we get a=1,b=−3√5 and c=10
Quadratic formula, x=−b±√b2−4ac2a,
Therefore, roots of given quadratic equation is given by
x=−(−3√5)±√(−3√5)2−4(1)(10)2(10)
=3√5±√45−402=3√5±√52
⇒x=3√5+√52 or x=3√5−√52
⇒x=2√5 or x=√5
Hence, the roots of the given equation are 2√5 and √5.
(vii) Given equation is 12x2−√11x+1=0
On comparing with ax2+bx+c=0, we get a=12,b=−√11 and c=1
Quadratic formula, x=−b±√b2−4ac2a,
Therefore, roots of given quadratic equation is given by
x=−(−√11)±√(−√11)2−4×12×12(12)
=√11±√11−2(1)
=√11±√9
⇒x=√11+3 or x=√11−3
Hence, the roots of the given equation are
√11+3 and √11−3.