Final answer:
To find the x-values for which the curve y = x³ + x² has a slope of 1, calculate the derivative of the function and set it equal to 1, then solve for x using the quadratic formula.
Step-by-step explanation:
To find the values of x for which y = x³ + x² has a slope of 1, we need to calculate the derivative of the function, which represents the slope of the tangent line at any point on the curve. The derivative of y = x³ + x² is y' = 3x² + 2x. We set this equal to 1 and solve for x:
3x² + 2x = 1
Rearranging yields 3x² + 2x - 1 = 0. This is a quadratic equation which can be solved using the quadratic formula, x = [-b ± sqrt(b² - 4ac)] / (2a). After completion, you will have the x-values for which the slope of the curve y = x³ + x² is 1.
Learn more about Mathematical Slopes