Question

If the difference of the roots of the quadratic equation is 5 and the difference of their cubes is 215, then the quadratic equation is $x^2\pm 7x+m=0$.
The value of $m$ is

• A. 2
• B. 4
• C. 6
• D. 10

Answer 1

The correct option is C 6

Let the roots be $a$ and $b$
$a^3-b^3=215$
$\Rightarrow a-b=5$

cubing both sides,
$(a-b{)}^3=125$
$\Rightarrow a^3-b^3-3ab(a-b)=125$
$\Rightarrow 215-3ab\left(5\right)=125$ (putting the values)

$\Rightarrow -3ab\left(5\right)=125-215$
$\Rightarrow -ab=-\frac{90}{15}$
$ab=6$ ....(1)
Now, $(a+b{)}^2=(a-b{)}^2+4ab$

$(a+b{)}^2=25+4\times 6$
$(a+b{)}^2=49$
$a+b=\pm 7$ ......(2)
Standard form of the equation is: $x^2-Sx+P=0$,

where $S$=sum of roots and $P$= Product of roots

From (1) and (2)
the equation can be written as:
$x^2\pm 7x+6=0$

Comparing with $x^2\pm 7x+m=0$

we get $m=6$