Question

which of the following statements best describes how to derive the quadratic formula? choose the best statement. \\( \bigcirc \\) a. the quadratic formula can be derived by solving the equation \\( ax^2 + bx + c = 0, a \
eq 0 \\), for \\( x \\) using the zero product property. \\( \bigcirc \\) b. the quadratic formula can be derived by solving the equation \\( ax^2 + bx + c = 0, a \
eq 0 \\), for \\( x \\) by factoring. \\( \bigcirc \\) c. the quadratic formula can be derived by solving the equation \\( ax^2 + bx + c = 0, a \
eq 0 \\), for \\( x \\) using the method of completing the square. \\( \bigcirc \\) d. the quadratic formula can only be derived using calculus.

Explanation:

• Option A: The zero - product property is used to solve factored quadratic equations (like \((x - r_1)(x - r_2)=0\) gives \(x = r_1\) or \(x = r_2\)), not to derive the quadratic formula.
• Option B: Factoring is only applicable to certain quadratic equations (those that can be factored into binomials with integer coefficients) and is not the method used to derive the general quadratic formula.
• Option C: The quadratic formula is derived by starting with the general quadratic equation \(ax^{2}+bx + c = 0,a

eq0\) and using the method of completing the square. We isolate the \(x\) terms, divide through by \(a\), complete the square on the \(x\) - related terms, and then solve for \(x\).

• Option D: The quadratic formula is an algebraic result and can be derived using algebraic methods (completing the square) without the use of calculus.

Answer:

C. The quadratic formula can be derived by solving the equation \(ax^{2}+bx + c = 0,a
eq0\), for \(x\) using the method of completing the square.