Question

If $a,b$ denote the distinct real roots of the quadratic equation $x^2+20x-2020=0$ and suppose $c,d$ denote the distinct complex roots of the quadratic equation $x^2+20x+2020=0.$ Find the value of $ac(a-c)+ad(a-d)+bc(b-c)+bd(b-d).$

• A. $-160800$
• B. $800$
• C. $16000$
• D. $20$

Answer 1

The correct option is A $-160800$

Given: $x^2+20x-2020=0$ with distinct real roots $a,b,$ and $x^2+20x+2020=0$ with distinct complex roots $c,d$

From $x^2+20x-2020,$
$a+b=-20,a\cdot b=-2020\cdots \left(i\right)$

From $x^2+20x+2020,$
$c+d=-20,c\cdot d=2020\cdots \left(ii\right)$

The expression we have to find is $ac(a-c)+ad(a-d)+bc(b-c)+bd(b-d)$
$\Rightarrow a^{2}c-a{c}^2+a^{2}d-a{d}^2+b^{2}c-b{c}^2+b^{2}d-b{d}^2$
$\Rightarrow a^2(c+d)-c^2(a+b)-d^2(a+b)+b^2(c+d)$
$\Rightarrow (a^2+b^2)(c+d)-(a+b)(c^2+d^2)$
We know that
$(x+y{)}^2=x^2+y^2+2xy$
$x^2+y^2=(x+y{)}^2-2xy$
Using this relation here
$\Rightarrow (c+d)\left[\right(a+b{)}^2-2ab]-(a+b\left)\right[(c+d{)}^2-2cd$
$\Rightarrow (-20)\left[\right(-20{)}^2-(2\times (-2020\left)\right)]-(-20\left)\right[(-20{)}^2-2\times 2020]$
$\Rightarrow (-20)[400+4040]+20[400-4040]$
$\Rightarrow 20[-400-4040+400-4040]$
$\Rightarrow -20\times 8040$
$\Rightarrow -160800$