Answer :
Final Answer:
The real solutions of the equation -19x³ - 12x² + 16x = 0 are x = 0 and x ≈ -0.529.
Explanation:
To find the real solutions of the given cubic equation -19x³ - 12x² + 16x = 0, we utilize graphing techniques. The first step is to set the equation equal to zero and factor out common terms, yielding -x(19x² + 12x - 16) = 0. This equation has two factors: x = 0 and the quadratic factor 19x² + 12x - 16.
To analyze the quadratic factor, we graph it and observe the points where the graph intersects the x-axis, representing the real solutions. Utilizing graphing software or a graphing calculator, we find that the quadratic factor has roots at approximately x ≈ -0.529 and x ≈ 0.450.
Upon examining the original cubic equation, we recognize that x = 0 is a solution. Additionally, from the quadratic factor, x ≈ -0.529 is another real solution. Therefore, the final answer is that the real solutions of the given cubic equation are x = 0 and x ≈ -0.529.
In summary, the graphing method enables us to identify the points where the equation intersects the x-axis, indicating the real solutions. This approach provides a visual representation of the roots, facilitating a clear understanding of the solutions to the cubic equation. Graphing software or calculators prove invaluable in efficiently determining these solutions without the need for complex algebraic manipulations.
