Question

What is the product of the roots of the equation `-2x^2+3x+8=0`?

Answer 1

-4

Explanation:

for any quadratic

`ax^2+bx+c=0" "`if the roots are `" "alpha" "&" "beta`

the sum of roots`" "alpha+beta=-b/a`

the product of the roots`" "alphabeta=c/a`

these results do need to be memorised.

**see proof below.

for `-2x^2+3x+8=0`

product of roots`" "aplhabeta=8/-2=-4`

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*proof of results

`ax^2+bx+c=0" "`if the roots are `" "alpha" "&" "beta`

`(x-alpha)(x-beta)=0`

expanding

`x^2-betax-alphax+alphabeta=0`

`x^2color(red)(-(alpha+beta))x+color(blue)(alphabeta)=0" "---(1)`

`ax^2+bx+c=0" "`

`-:a`

`=>x^2+color(red)((b/a))x+color(blue)(c/a)=0" "---(2)`

comparing `""--(1)" "&""--(2)" "`the results follow.