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

Dec 17, 2016

## -4

#### Explanation:

$a {x}^{2} + b x + c = 0 \text{ }$if the roots are $\text{ "alpha" "&" } \beta$

the sum of roots$\text{ } \alpha + \beta = - \frac{b}{a}$

the product of the roots$\text{ } \alpha \beta = \frac{c}{a}$

these results do need to be memorised.

**see proof below.

for $- 2 {x}^{2} + 3 x + 8 = 0$

product of roots$\text{ } a p l h a \beta = \frac{8}{-} 2 = - 4$

*proof of results

$a {x}^{2} + b x + c = 0 \text{ }$if the roots are $\text{ "alpha" "&" } \beta$

$\left(x - \alpha\right) \left(x - \beta\right) = 0$

expanding

${x}^{2} - \beta x - \alpha x + \alpha \beta = 0$

${x}^{2} \textcolor{red}{- \left(\alpha + \beta\right)} x + \textcolor{b l u e}{\alpha \beta} = 0 \text{ } - - - \left(1\right)$

$a {x}^{2} + b x + c = 0 \text{ }$

$\div a$

$\implies {x}^{2} + \textcolor{red}{\left(\frac{b}{a}\right)} x + \textcolor{b l u e}{\frac{c}{a}} = 0 \text{ } - - - \left(2\right)$

comparing $\text{--(1)" "&""--(2)" }$the results follow.