Question

If a, b, c and d are the roots of the equation

2x⁴-4x³+6x²+10x+12=0.

Find the value of a²+b²+c²+d².

Answer 1

$$(a+b+c+d)^2$$

$$=(a+b)^2+2(a+b)(c+d)+(c+d)^2$$

$$=a^2+2ab+b^2+2ac+2ad+2bc+2bd$$

$$+c^2+2cd+d^2$$

Then

$$a^2+b^2+c^2+d^2$$

$$=(a+b+c+d)^2$$

$$-2(ab+ac+ad+bc+bd+cd)$$

Use Vieta's formulas

$$a+b+c+d=-\dfrac{-4}{2}=2$$

$$ab+ac+ad+bc+bd+cd=\dfrac{6}{2}=3$$

Hence

$$a^2+b^2+c^2+d^2=(2)^2-2(3)=-2$$