Section 9.4 Quadratic Formula
Example 9.4.1.
\begin{align*}
ax^2 + bx + c = 0 \amp \quad \text{ Separate constant from variables } \\
-c\ -c \amp\quad \text{ Subtract $c$ from both sides } \\
ax^2 + bx =- c \amp\quad \text{ Divide each term by } a \\
\frac{ax^2}{a} + \frac{bx}{a} =\frac{- c}{a} \amp\quad \text{ } \\
x^2+\frac{b}{a}x=\frac{-c}{a} \amp \quad \text{ Find the number that completes the square } \\
\left(\frac{1}{2}\cdot\frac{b}{a}\right)^2=\left(\frac{b}{2a}\right)^2=\frac{b^2}{4a^2} \amp\quad \text{ Add to both sides } \\
\frac{b^2}{4a^2}-\frac{c}{a}\left(\frac{4a}{4a}\right)=\frac{b^2}{4a^2}-\frac{4ac}{4a^2}=\frac{b^2-4ac}{4a^2} \amp\quad \text{ Get common denominator on right } \\
x^2+\frac{b}{a}x+\frac{b^2}{4a^2}=\frac{b^2}{4a^2}-\frac{4ac}{4a^2}=\frac{b^2-4ac}{4a^2} \amp\quad \text{ Factor } \\
\left(x+\frac{b}{2a}\right)^2= \frac{b^2-4ac}{4a^2}\amp \quad \text{ Solve using the even root property } \\
\sqrt{\left(x+\frac{b}{2a}\right)^2}= \pm\sqrt{\frac{b^2-4ac}{4a^2}} \amp\quad \text{ Simplify roots } \\
x+\frac{b}{2a}=\frac{\pm\sqrt{b^2-4ac}}{2a} \amp\quad \text{ Subtract $\frac{b}{2a}$ from both sides } \\
\amp\quad \text{ } \\
x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \amp\quad \text{ Our Solution }
\end{align*}
\begin{equation*}
\textbf{ Quadratic Formula: if } \mathbf{ax^2 +bx+c=0} \textbf{ then } \ \mathbf{ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} }
\end{equation*}
Example 9.4.2.
\begin{align*}
x^2 +3x+2=0 \amp \quad \text{ $a=1$, $b=3$, $c=2$, use quadratic formula } \\
x=\frac{-3\pm\sqrt{3^2-4(1)(2)}}{2(1)} \amp\quad \text{ Evaluate exponent and multiplication } \\
x=\frac{-3\pm\sqrt{9-8}}{2} \amp\quad \text{ Evaluate subtraction under root } \\
x=\frac{-3\pm\sqrt{1}}{2} \amp\quad \text{ Evaluate root } \\
x=\frac{-3\pm 1}{2} \amp\quad \text{ Evaluate $\pm$ to get two answers } \\
x=\frac{-2}{2} \text{ or } \frac{-4}{2} \amp\quad \text{ Simplify fractions } \\
x=-1 \text{ or } -2 \amp\quad \text{ Our Solution }
\end{align*}
Example 9.4.3.
\begin{align*}
25x^2 = 30x + 11 \amp \quad \text{ First set equal to zero } \\
- 30x - 11\ - 30x - 11 \amp \quad \text{ Subtract $30x$ and $11$ from both sides } \\
25x^2-30x-11=0 \amp \quad \text{ $a=25$, $b=-30$, $c=-11$, use quadratic formula } \\
x=\frac{30\pm\sqrt{(-30)^2-4(25)(-11)}}{2(25)} \amp\quad \text{ Evaluate exponent and multiplication } \\
x=\frac{30\pm\sqrt{900+1100}}{50} \amp\quad \text{ Evaluate subtraction under root } \\
x=\frac{30\pm\sqrt{2000}}{50} \amp\quad \text{ Evaluate root } \\
x=\frac{30\pm 20\sqrt{5}}{50} \amp\quad \text{ Reduce fraction by dividing each term by } 10 \\
x=\frac{3\pm 2\sqrt{5}}{5} \amp\quad \text{ Our Solution }
\end{align*}
Example 9.4.4.
\begin{align*}
3x^2+4x+8=2x^2+6x-5 \amp \quad \text{ First set equal to zero } \\
-2x^2 -6x+5 \ -2x^2 -6x+5 \amp \quad \text{ Subtract $2x^2$ and $6x$ and add $5$ } \\
x^2 -2x+13=0 \amp \quad \text{ $a=1$, $b=-2$, $c=13$, use quadratic formula } \\
x=\frac{2\pm\sqrt{(-2)^2-4(1)(13)}}{2(1)} \amp\quad \text{ Evaluate exponent and multiplication } \\
x=\frac{2\pm\sqrt{4-52}}{2} \amp\quad \text{ Evaluate subtraction under root } \\
x=\frac{2\pm\sqrt{-48}}{2} \amp\quad \text{ Simplify root } \\
x=\frac{2\pm 4i\sqrt{3}}{2} \amp\quad \text{ Reduce fraction by dividing each term by } 2 \\
x=1\pm2i\sqrt{3} \amp\quad \text{ Our Solution }
\end{align*}
Example 9.4.5.
\begin{align*}
4x^2 -12x+9=0 \amp \quad \text{ $a=4$, $b=-12$, $c=9$, use quadratic formula } \\
x=\frac{12\pm\sqrt{(-12)^2-4(4)(9)}}{2(4)} \amp\quad \text{ Evaluate exponents and multiplication
} \\
x=\frac{12\pm\sqrt{144-144}}{8} \amp\quad \text{ Evaluate subtraction inside root } \\
x=\frac{12\pm\sqrt{0}}{8} \amp\quad \text{ Evaluate root } \\
x=\frac{12\pm 0}{8} \amp\quad \text{ Evaluate } \pm \\
x=\frac{12}{8} \amp\quad \text{ Reduce fraction } \\
x=\frac{3}{2} \amp\quad \text{ Our Solution }
\end{align*}
Example 9.4.6.
\begin{align*}
3x^2 +7=0 \amp \quad \text{ $a=3$, $b=0$ (missing term), } c=7 \\
x=\frac{-0\pm\sqrt{0^2-4(3)(7)}}{2(3)} \amp\quad \text{ Evaluate exponnets and multiplication, zeros not needed } \\
x=\frac{\pm\sqrt{-84}}{6} \amp\quad \text{ Simplify root } \\
x=\frac{\pm 2i\sqrt{21}}{6} \amp\quad \text{ Reduce, dividing by } 2 \\
x=\frac{\pm i\sqrt{21}}{3} \amp\quad \text{ Our Solution }
\end{align*}
