If b and care constants and x ^ 2 + bx + c = (x + 3)(x - 5) for all values of x, what is the value of b?

Rationale

b = -2.

To determine the value of b in the equation (x^2 + bx + c = (x + 3)(x - 5)), we first expand the right-hand side. The expansion results in a standard quadratic form, from which we can directly compare coefficients with the left-hand side.

A (-5) YOUR ANSWER

This choice arises from the constant term in the factors (x + 3) and \(x - 5\). However, -5 is not the value of b; it is actually the negative of one of the roots from the factorization, making it incorrect for our comparison of coefficients.

B (-2) CORRECT

This is the correct answer because when we expand ((x + 3)(x - 5)), we find (x^2 - 2x - 15). In this expanded form, the coefficient of (x) (which corresponds to b) is indeed -2, confirming that this choice is accurate.

C (3) YOUR ANSWER

Choosing 3 suggests a misunderstanding of the relationship between the coefficients and the roots of the quadratic equation. While 3 is present in the factor (x + 3), it does not correspond to the coefficient b in the standard form of the quadratic equation.

D (8) YOUR ANSWER

This option does not appear in the expansion of ((x + 3)(x - 5)). Instead, it misrepresents the linear coefficient of the resulting quadratic expression, which is -2, thus making this choice incorrect.

E (15) YOUR ANSWER

This choice represents the constant term resulting from the multiplication of the two factors but does not relate to the coefficient b. The constant term does not affect the value of b in the equation, which is required for the linear term.

Conclusion

Identifying the value of b involves expanding the factored form of the quadratic equation and equating coefficients. The correct coefficient of x from the expansion of ((x + 3)(x - 5)) yields b = -2, while the other options misinterpret the relationships within the quadratic expression. Understanding this relationship is crucial for correctly solving similar algebraic equations.

If b and care constants and x ^ 2 + bx + c = (x + 3)(x - 5) for all values of x, what is the value of b?

b = -2.

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