Which equation shows the quadratic formula used correctly to solve $a{x}^2+bx+c=0$ for $x$?

• A. $x=\frac{-b\pm \sqrt{b^2-4ac}}{2c}$
• B. $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$
• C. $x=\frac{-b\pm \sqrt{b^2+4ac}}{2a}$
• D. $x=\frac{b\pm \sqrt{b^2-4ac}}{2a}$

Answer: (B)

Verified step by step guidance

1. Recall the standard form of a quadratic equation: \(a x^{2} + b x + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a \neq 0\).
2. The quadratic formula is used to find the solutions for \(x\) in the quadratic equation and is derived from completing the square on the general form.
3. The correct quadratic formula is given by: \[x = \frac{-b \pm \sqrt{b^{2} - 4 a c}}{2 a}\]
4. Note that the denominator must be \$2a\(, not \)2c\(, because it comes from the coefficient of \)x^{2}$ in the original equation.
5. Also, the expression under the square root (the discriminant) is \(b^{2} - 4 a c\), not \(b^{2} + 4 a c\), as it determines the nature of the roots.