describe how to derive the quadratic formula from a quadratic equation in standard form.

Explanation:

Step1: Start with standard form

Let the quadratic equation be $ax^{2}+bx + c=0$, where $a\neq0$. First, divide the entire equation by $a$: $x^{2}+\frac{b}{a}x+\frac{c}{a}=0$.

Step2: Isolate the $x$ - terms

Subtract $\frac{c}{a}$ from both sides: $x^{2}+\frac{b}{a}x=-\frac{c}{a}$.

Step3: Complete the square

Take half of the coefficient of $x$, square it and add it to both sides. The coefficient of $x$ is $\frac{b}{a}$, half of it is $\frac{b}{2a}$, and its square is $\frac{b^{2}}{4a^{2}}$. So we have $x^{2}+\frac{b}{a}x+\frac{b^{2}}{4a^{2}}=-\frac{c}{a}+\frac{b^{2}}{4a^{2}}$.

Step4: Factor the left - hand side

The left - hand side is a perfect square: $(x + \frac{b}{2a})^{2}=-\frac{c}{a}+\frac{b^{2}}{4a^{2}}$. Combine the right - hand side terms: $(x+\frac{b}{2a})^{2}=\frac{b^{2}-4ac}{4a^{2}}$.

Step5: Take the square root of both sides

$x+\frac{b}{2a}=\pm\sqrt{\frac{b^{2}-4ac}{4a^{2}}}=\pm\frac{\sqrt{b^{2}-4ac}}{2a}$.

Step6: Solve for $x$

Subtract $\frac{b}{2a}$ from both sides: $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$.

Answer:

The quadratic formula $x = \frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ is derived from the standard - form quadratic equation $ax^{2}+bx + c = 0$ ($a\neq0$) by dividing by $a$, isolating the $x$ terms, completing the square, factoring, taking the square root, and solving for $x$.