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Quadratics


The quadratic formula:
\(\frac{-b\pm \sqrt{b^2-4ac}}{2a}\)
If \( ax^2 + bx + c = 0 \), where \( a \ne 0 \), then: \[x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}\]

Solve the following equations using the quadratic formula
\[ (1) \quad x^2 -13 x + 22 = 0 \]\[ (2) \quad 4 x^2 + 6 x -10 = 0 \]\[ (3) \quad -2 x^2 -11 x + 6 = 0 \]\[ (4) \quad x^2 -2 x -99 = 0 \]\[ (5) \quad 3 x^2 + 8 x + 5 = 0 \]

Example (1)


\[ x^2 -13 x + 22 = 0 \]\[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \]\[ x = \frac{-(-13) \pm \sqrt{(-13)^2-4(1)(22)}}{2(1)} \]\[ x = \frac{ 13 \pm \sqrt{ 81 }}{ 2 } \]\[ x = \frac{ 13 \pm 9}{ 2 } \]
\begin{align} x_1 & = \frac{ 13 - 9}{ 2 } \\ & = \frac{ 4}{ 2 } \\ & = 2 \end{align}\begin{align} x_2 & = \frac{ 13 + 9}{ 2 } \\ & = \frac{ 22}{ 2 } \\ & = 11 \end{align}
\[ \therefore x = 2 \quad \text{or} \quad 11 \]

Example (2)


\[ 4 x^2 + 6 x -10 = 0 \]\[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \]\[ x = \frac{-(6) \pm \sqrt{(6)^2-4(4)(-10)}}{2(4)} \]\[ x = \frac{ -6 \pm \sqrt{ 196 }}{ 8 } \]\[ x = \frac{ -6 \pm 14}{ 8 } \]
\begin{align} x_1 & = \frac{ -6 - 14}{ 8 } \\ & = \frac{ -20}{ 8 } \\ & = - \frac{ 5}{ 2 } \end{align}\begin{align} x_2 & = \frac{ -6 + 14}{ 8 } \\ & = \frac{ 8}{ 8 } \\ & = 1 \end{align}
\[ \therefore x = - \frac{ 5}{ 2 } \quad \text{or} \quad 1 \]

Example (3)


\[ -2 x^2 -11 x + 6 = 0 \]\[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \]\[ x = \frac{-(-11) \pm \sqrt{(-11)^2-4(-2)(6)}}{2(-2)} \]\[ x = \frac{ 11 \pm \sqrt{ 169 }}{ -4 } \]\[ x = \frac{ 11 \pm 13}{ -4 } \]
\begin{align} x_1 & = \frac{ 11 - 13}{ -4 } \\ & = \frac{ -2}{ -4 } \\ & = \frac{ 1}{ 2 } \end{align}\begin{align} x_2 & = \frac{ 11 + 13}{ -4 } \\ & = \frac{ 24}{ -4 } \\ & = -6 \end{align}
\[ \therefore x = \frac{ 1}{ 2 } \quad \text{or} \quad -6 \]

Example (4)


\[ x^2 -2 x -99 = 0 \]\[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2-4(1)(-99)}}{2(1)} \]\[ x = \frac{ 2 \pm \sqrt{ 400 }}{ 2 } \]\[ x = \frac{ 2 \pm 20}{ 2 } \]
\begin{align} x_1 & = \frac{ 2 - 20}{ 2 } \\ & = \frac{ -18}{ 2 } \\ & = -9 \end{align}\begin{align} x_2 & = \frac{ 2 + 20}{ 2 } \\ & = \frac{ 22}{ 2 } \\ & = 11 \end{align}
\[ \therefore x = -9 \quad \text{or} \quad 11 \]

Example (5)


\[ 3 x^2 + 8 x + 5 = 0 \]\[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \]\[ x = \frac{-(8) \pm \sqrt{(8)^2-4(3)(5)}}{2(3)} \]\[ x = \frac{ -8 \pm \sqrt{ 4 }}{ 6 } \]\[ x = \frac{ -8 \pm 2}{ 6 } \]
\begin{align} x_1 & = \frac{ -8 - 2}{ 6 } \\ & = \frac{ -10}{ 6 } \\ & = - \frac{ 5}{ 3 } \end{align}\begin{align} x_2 & = \frac{ -8 + 2}{ 6 } \\ & = \frac{ -6}{ 6 } \\ & = -1 \end{align}
\[ \therefore x = - \frac{ 5}{ 3 } \quad \text{or} \quad -1 \]