\begin{definition}[Quadratic Formula] The quadratic formula is a formula that provides the solutions to a quadratic equation in standard form \$ax^2 + bx + c = 0\$:
\$\$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\$\$
The solutions are given by the two values:
- \$x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a}\$
- \$x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a}\$ \end{definition}
The quadratic formula is a universal method for solving quadratic equations, as it always works (even when the quadratic is not factorable).
\begin{note} The quadratic formula is derived from the process of completing the square. \end{note}
\begin{proof} The quadratic formula can be derived by completing the square on the general quadratic equation \$ax^2 + bx + c = 0\$:
- Start with the general quadratic equation:
\$\$ax^2 + bx + c = 0\$\$
- Move the constant term to the other side:
\$\$ax^2 + bx = -c\$\$
- Divide the entire equation by \$a\$ to make the coefficient of \$x^2\$ equal to 1:
\$\$x^2 + \frac{b}{a}x = -\frac{c}{a}\$\$
- Complete the square by adding \$\left(\frac{b}{2a}\right)^2\$ to both sides:
\$\$x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2\$\$
- The left side is now a perfect square trinomial:
\$\$ \left(x + \frac{b}{2a}\right)^2 = -\frac{c}{a} + \frac{b^2}{4a^2}\$\$
- Simplify the right side:
\$\$ \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}\$\$
- Take the square root of both sides:
\$\$x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a}\$\$
- Solve for \$x\$ by subtracting \$\frac{b}{2a}\$ from both sides:
\$\$x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a}\$\$
- Combine the terms on the right side:
\$\$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\$\$
This is the quadratic formula. \end{proof}
\begin{note} The quadratic formula provides the solutions to the quadratic equation \$ax^2 + bx + c = 0\$ as long as the discriminant \$b^2 - 4ac\$ is non-negative.
If the discriminant is negative, the quadratic equation has no real solutions. \end{note}
\begin{warning} The quadratic formula only provides the solutions to the quadratic equation \$ax^2 + bx + c = 0\$ if the equation is equal to zero.
If the equation is not equal to zero, you must first rearrange it so that it is equal to zero. \end{warning}
\begin{example} Solve the quadratic equation \$2x^2 - 3x - 5 = 0\$ using the quadratic formula.
- Identify the coefficients \$a\$, \$b\$, and \$c\$:
\$\$a = 2, \quad b = -3, \quad c = -5\$\$
- Substitute these values into the quadratic formula:
\$\$x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 2 \cdot (-5)}}{2 \cdot 2}\$\$
- Simplify the expression:
\$\$x = \frac{3 \pm \sqrt{9 + 40}}{4}\$\$
- Further simplify:
\$\$x = \frac{3 \pm \sqrt{49}}{4}\$\$
- Calculate the two possible solutions:
\$\$x_1 = \frac{3 + 7}{4} = \frac{10}{4} = 2.5\$\$
\$\$x_2 = \frac{3 - 7}{4} = \frac{-4}{4} = -1\$\$
The solutions are \$x = 2.5\$ and \$x = -1\$. \end{example}
\begin{self_review} Solve the following quadratic equations using the quadratic formula:
- \$x^2 + 4x + 4 = 0\$
- \$3x^2 - 2x - 1 = 0\$
- \$2x^2 + 3x - 5 = 0\$ \end{self_review}
\begin{tok} The quadratic formula is a universal method for solving quadratic equations. However, it is not always the most efficient method. In some cases, factoring or completing the square may be faster.
How do we decide which method to use? Is it always better to use the most efficient method, or is it sometimes better to use a universal method that always works? \end{tok}
