1. The problem states: For the quadratic equation $$3.089\cdot 2x^1 - (1 - 3b)x + 5b = 0$$ one root is $b$. We need to find the value of $b$. 2. The general form of a quadratic equation is $$ax^2 + bx + c = 0$$ and if one root is known, say $r$, then it satisfies the equation: $$ar^2 + br + c = 0$$. 3. Here, the equation is $$3.089\cdot 2x - (1 - 3b)x + 5b = 0$$. Simplify the coefficient of $x$: $$3.089 \times 2x = 6.178x$$ So the equation becomes: $$6.178x - (1 - 3b)x + 5b = 0$$ 4. Combine like terms for $x$: $$6.178x - x + 3bx + 5b = 0$$ $$ (6.178 - 1 + 3b)x + 5b = 0$$ $$ (5.178 + 3b)x + 5b = 0$$ 5. Since $b$ is a root, substitute $x = b$: $$ (5.178 + 3b) b + 5b = 0$$ $$ 5.178b + 3b^2 + 5b = 0$$ 6. Combine like terms: $$3b^2 + (5.178 + 5) b = 0$$ $$3b^2 + 10.178b = 0$$ 7. Factor out $b$: $$b(3b + 10.178) = 0$$ 8. Set each factor equal to zero: - $b = 0$ - $3b + 10.178 = 0 \Rightarrow b = -\frac{10.178}{3} \approx -3.3927$ 9. Check the options given: A. -0.4; 0 B. -2.5; 0 C. 2.5; 0 D. 0 E. -0.8; 0 Only option D matches $b=0$ exactly. Final answer: $b = 0$