Answer:
8x² +36x -27=0
Step-by-step explanation:
2x²= 6x +3
Let's rewrite the equation into the form of ax²+bx+c= 0.
2x² -6x -3=0
Thus, a= 2
b= -6
c= -3
Sum of roots= [tex] - \frac{b}{a} [/tex]
Since your roots are p and q, sum of roots= p +q
p +q= [tex] - (\frac{ - 6}{2} )[/tex]
p +q= 6 ÷2
p +q= 3
Product of roots= [tex] \frac{c}{a} [/tex]
pq= [tex] - \frac{3}{2} [/tex]
Quadratic equations:
x² -(sum of roots)x +(product of roots)= 0
Thus, we have to find the sum and the product of the new roots, p²q and pq².
p +q= 3
pq= -3/2
Product of new roots
= (p²q)(pq²)
= p³q³
= (pq)³
[tex] = ( - \frac{3}{2} )^{3} \\ = - \frac{27}{8} [/tex]
sum of new roots
= p²q +pq²
= pq(p +q)
= (-3/2)(3)
= -9/2
Thus, the quadratic equation with roots p²q and pq² is
x² -(-9/2)x -27/8 = 0
[tex]x ^{2} + \frac{9}{2} x - \frac{27}{8} = 0[/tex]
Multiply by 8 throughout:
[tex]8 {x}^{2} + 36x - 27 = 0[/tex]
