The sum of squares of the roots of x2 + 8x + 15 = 0 is …………A)

1.	The sum of squares of the roots of x2 + 8x + 15 = 0 is …………A) 30 B) 34 C) 40 D) 44
Answer» Correct option is (B) 34 Given quadratic equation is \(x^2+8x+15=0\) Let roots are \(\alpha\;and\;\beta.\) \(\therefore\) Sum of roots \(=\frac{-8}1=-8\) \(\Rightarrow\) \(\alpha+\beta\) = -8 _____________(1) And product of roots \(=\frac{15}1=15\) \(\Rightarrow\) \(\alpha\beta=15\) _____________(2) Now, \((\alpha-\beta)^2=(\alpha+\beta)^2-4\alpha\beta\) \(=(-8)^2-4\times15\) = 64 - 60 = 4 \(=2^2\) \(\therefore\) \(\alpha-\beta=2\) _____________(3) By adding equations (1) & (3), we get \((\alpha+\beta)+(\alpha-\beta)=-8+2\) \(\Rightarrow\) \(2\alpha=-6\) \(\Rightarrow\) \(\alpha=\frac{-6}2=-3\) \(\therefore\) \(\beta=-8-\alpha\) (From (1)) = -8 - (-3) = -8+3 = -5 \(\therefore\) \(\alpha^2+\beta^2=(-3)^2+(-5)^2\) = 9+25 = 34 Alternative :- \(\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta\) \(=(-8)^2-2\times15\) = 64 - 30 = 34 Correct option is B) 34

The sum of squares of the roots of x2 + 8x + 15 = 0 is …………A) 30 B) 34 C) 40 D) 44

Answer»

Correct option is (B) 34

Given quadratic equation is \(x^2+8x+15=0\)

Let roots are \(\alpha\;and\;\beta.\)

\(\therefore\) Sum of roots \(=\frac{-8}1=-8\)

\(\Rightarrow\) \(\alpha+\beta\) = -8 _____________(1)

And product of roots \(=\frac{15}1=15\)

\(\Rightarrow\) \(\alpha\beta=15\) _____________(2)

Now, \((\alpha-\beta)^2=(\alpha+\beta)^2-4\alpha\beta\)

\(=(-8)^2-4\times15\)

= 64 - 60

= 4 \(=2^2\)

\(\therefore\) \(\alpha-\beta=2\) _____________(3)

By adding equations (1) & (3), we get

\((\alpha+\beta)+(\alpha-\beta)=-8+2\)

\(\Rightarrow\) \(2\alpha=-6\)

\(\Rightarrow\) \(\alpha=\frac{-6}2=-3\)

\(\therefore\) \(\beta=-8-\alpha\) (From (1))

= -8 - (-3)

= -8+3 = -5

\(\therefore\) \(\alpha^2+\beta^2=(-3)^2+(-5)^2\)

= 9+25 = 34

Alternative :-

\(\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta\)

\(=(-8)^2-2\times15\)

= 64 - 30 = 34

Correct option is B) 34