Question

if p and q are roots of equation x^2 +mx + m^2 +a = 0, then value of p ^2 +q^2 + pq is

Answer 1

Concept

The relationship between roots and coefficients is as follows: in a quadratic equation, the sum of the roots is equal to the negative value of the coefficient at the second term, divided by the coefficient at the first term. The product of the roots is equal to the third term divided by the first term.

Given

We have given an equation $x^2+mx+m^2+a=0$ and its roots which are p and q .

Find

We are asked to determine the value of  $p^2+q^2+pq$ .

Solution

It is given that p and q are the roots of the given equation $x^2+mx+m^2+a=0$

$a=1, b=m ,c=m^2+a$

The sum of roots $=\frac{-b}{a}$

       $p+q=\frac{-m}{1}\\p+q=-m$

Squaring on both sides, we get

$(p+q)^2=(-m)^2\\p^2+q^2+2pq=m^2$ .....(1)

Product of roots $=\frac{c}{a}$

        $pq=\frac{m^2+a}{1} \\\\pq=m^2+a$ .....(2)

Subtracting equation (2) from equation (1) , we get

$p ^2 +q^2 + 2pq-pq=m^2-(m^2+a)\\p ^2 +q^2 + pq=m^2-m^2-a\\p ^2 +q^2 + pq=-a$

Therefore, the value of equation $p^2+q^2+pq$ is -a .

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Answer 2

Answer:

The value of $p^2+q^2+pq$ is $-a$.

Step-by-step explanation:

Given: $p$ and $q$ are the roots of the equation $x^2+mx+m^2+a=0$

To find the value of $p^2+q^2+pq$.

Step 1 of 2

Consider the equation as follows:

$x^2+mx+(m^2+a)=0$

Here, $a=1$, $b=m$ and $c=m^2+a$.

Then,

The sum of the roots = $-\frac{b}{a}$

⇒ $p+q=-\frac{b}{a}$

⇒ $p+q=-\frac{m}{1}$

⇒ $p+q=-m$ . . . . . (1)

The product of the roots = $\frac{c}{a}$

⇒ $pq=\frac{c}{a}$

⇒ $pq=\frac{m^2+a}{1}$

⇒ $pq=m^2+a$ . . . . . (2)

Step 2 of 2

Find the value of $p^2+q^2+pq$.

From (1), we have

$p+q=-m$

Squaring both the sides as follows:

⇒ $(p+q)^2=(-m)^2$

⇒ $p^2+q^2+2pq=m^2$

⇒ $p^2+q^2+pq+pq=m^2$

⇒ $p^2+q^2+pq=m^2-pq$

⇒ $p^2+q^2+pq=m^2-(m^2+a)$ (From (2))

⇒ $p^2+q^2+pq=-a$

Final answer: The value of $p^2+q^2+pq$ is $-a$.

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