Question

First, see if you can cancel factors in the numerator and denominator. Then, using factors from the denominator make an equation stating that the fraction = Ax+B/(quadfactor₁) + C/(factor₂) where A, B, and C are constants. Multiply both sides by the denominator to give the numerator = Ax+B(factor₂) + C(quadfactor), then by substituting x = root₂, we can solve for X, then substitute x = 0, and use the new value for C to evaluate A and B [for A it might be easier to equate the x² coefficient in the original fraction].

a) Simplify the expression
b) Solve for the constant A
c) Find the value of B
d) Determine the value of C

Answer 1

The question involves simplifying a quadratic expression and finding constants in partial fraction decomposition. This mathematical process includes checking for factors to cancel, applying partial fraction techniques, and solving for unknowns by substituting specific x values.

Step-by-step explanation:

The student's question asks for help with solving for constants A, B, and C in the partial fraction decomposition of a rational expression, and ultimately finding the simplified form of that expression. This is a calculus or higher-level algebra topic typically covered in high school or early college courses. The given equation and additional context suggest that the starting point is a quadratic equation of the form ax² + bx + c = 0.

To simplify a complex fraction or solve for constants in the partial fraction decomposition:

1. Begin by checking if factors in the numerator and denominator can be canceled.
2. Apply the partial fraction decomposition technique to express the fraction as a sum of simpler fractions with unknown constants A, B, and C.
3. Multiply through by the common denominator to eliminate the fractions.
4. Substitute specific values of x to solve for the unknown constants.
5. Replace the found constants into the original equation to see if the original expression can be simplified further or to check the correctness of the solution.

For example, if given a simplified quadratic equation such as x² + 0.0211x - 0.0211 = 0, you would follow the steps outlined above to find the constants and simplify the expression. The quadratic formula, which is x = √(4ac), could be used to find the roots of the quadratic equation as well.