1.

Find the roots of following quadratic equations by using quadratic formula, if they exist. (i)2x^(2)+x-4=0 (ii) 2x^(2)+x+4=0 (iii) 2x^(2)+5sqrt3x+6=0 (iv) sqrt3x^(2)+11x+6sqrt3=0

Answer»

Solution :(i) GIVEN equation is
`2x^(2)+x-4=0`
) On comparing with `ax^(2)+bx+c=0`, we get
a=2,b=1 and c=-4
`:. "DISCRIMINAT", D=b^(2)-4ac` `impliesD=(1)^(2)-4xx2xx(-4)`
`impliesD=1+32`
`impliesD=33gt0`
Hence, the given equation has two real roots.
`:.x=(-b+-sqrtD)/(2a)`
`impliesx=(-1+-sqrt33)/(4)`
`impliesx=(-1+sqrt33)/(4),(-1-sqrt33)/(4)` are roots of to the equation.
(ii) Give equation is `2x^(2)+x+4=0`
On comparing with `ax^(2)+bx+c=0`, we get a=2, b=1 and c=4
`:."Discriminant",D=b^(2)-4ac`
`impliesD=(1)^(2)-4xx2xx4`
`impliesD=1-32`
`impliesD=-31lt0`
Hence, the equation has no real roots.
(iii) Given equation is `2x^(2)+5sqrt3x+6=0`
On comparing with `ax^(2)+bx+c=0`, we get
`a=2,b=5sqrt3andc=6`
`:."Discriminant"D=b^(2)-4ac`
`impliesD=(5sqrt3)^(2)-4xx2xx6`
`impliesD=75-48`
`impliesD=27gt0`
Hence, the equation has two real roots.
`:.x=(-b+-sqrtD)/(2a)`
`impliesx=(-5sqrt3+-sqrt27)/(4)impliesx=(-5sqrt3+-3sqrt3)/(4)`
`impliesx=(-2sqrt3)/(2)and(-8sqrt3)/(4)`
`impliesx=(-sqrt3)/(2)and-2sqrt3` are roots of the equation.
(IV) Given equation is `sqrt3x^(2)+11x+6sqrt3=0`
On comparing with `ax^(2)+bx+c=0` we get
`a=sqrt3,b=11andc=6sqrt3`
`:."Discriminant"D=b^(2)-4ac`
`impliesD=11^(2)-4xxsqrt3xx6sqrt3`
`impliesD=121-72`
`impliesD=49gt0`
Hence, the given equation has two real roots.
`:.x=(-b+-sqrtD)/(2a)`
`impliesx=(-11+-sqrt49)/(2sqrt3)impliesx=(-11+-7)/(2sqrt3)`
`impliesx=(-11+7)/(2sqrt3)and(-11-7)/(2sqrt3)`
`impliesx=(-4)/(2sqrt3)xx(sqrt3)/(sqrt3)and(-18)/(2sqrt3)xx(sqrt3)/(sqrt3)`
`impliesx=(-4sqrt3)/(6)and(-18sqrt3)/(6)`
`impliesx=(-2sqrt3)/(3)and-3sqrt3` are roots of the equation.


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