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Question

The difference between the two roots of the equation 3x2+10x+c=0 is 423 .

Find the solutions for the equation.


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Solution

Finding solutions to the equation 3x2+10x+c=0:

Step-1: Finding two roots with the help of a formula.

Knowing the square roots of equations ax2+bx+c=0,is given by,

x=-b±b2-4ac2a …(I)

Comparing 3x2+10x+c=0 with ax2+bx+c=0 and getting

a=3,b=10,c=c

x=-10±102-4×3×c2×3=-10±100-12c6=-10±4×25-4×3×c6=-10±4×(25-3×c)6=-2×5±2(25-3×c)2×3=-5±(25-3×c)3

The square roots of the equation are x1=-5+(25-3×c)3,x2=-5-(25-3×c)3. …(II)

Step-2: Finding the value of c.

Given, The difference between two roots is 423 , that is, 143. From (II),

x1-x2=143-5+(25-3×c)3--5-(25-3×c)3=143-5+(25-3×c)+5+(25-3×c)3=1432(25-3×c)3=1432(25-3×c)=14(25-3×c)=7

Now squaring both sides,

(25-3×c)=4925-49=3c-24=3c-8=c

The value of c is -8.

Step-3: Finding solutions for the given equation.

Putting value of c in (II),

x1=-5+(25-(3×-8))3=-5+(25-(-24))3=-5+(25+24)3=-5+493=-5+73=23

x2=-5-(25-(3×-8))3=-5-(25-(-24))3=-5-(25+24)3=-5-493=-5-73=-123=-4

Hence, the solutions for the equation 3x2+10x+c=0 are 23,-4.


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