Find The Tree (3) Roots

Algebra Level 5

Find the product of the real roots of the following equation:

x 3 − 0.5 x 2 − 3.5 x − 6 2 x − 6 − 4 x − x 2 x − 3 = 0 \large \frac{x^3-0.5x^2-3.5x-6}{2x-6}-\frac{4x-x^2}{x-3}=0

2 -3 6 There are an infinite number of roots 4 There are no roots 3 0

Clive Chen by Clive Chen , 25, USA

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1 solution

f ( x ) = x 3 − . 5 x 2 − 3.5 x − 12 − 8 x + 2 x 2 2 x − 6 = 0. ⟹ f ( x ) = x 3 + 1.5 x 2 − 11.5 x − 6 2 x − 6 = 0. ∴ f ( x ) = 1 4 ∗ ( 2 x − 6 ) ( 2 x 2 + 9 x + 4 ) 2 x − 6 = 0. ∴ f ( x ) = ( 2 x + 1 ) ( x + 4 ) = 0. O N L Y t w o r o o t s . f(x)=\dfrac{x^3-.5x^2-3.5x-12-8x+2x^2}{2x-6}=0.\\ \implies\ f(x)=\dfrac{x^3+1.5x^2-11.5x-6}{2x-6}=0.\\ \therefore\ f(x)=\dfrac{\frac 1 4 *(2x-6)(2x^2+9x+4)}{2x-6}=0.\\ \therefore\ f(x)=(2x+1)(x+4)=0.\\ ONLY\ \ two\ \ roots.

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