Incomplete quadratic equations II
Let's draw two incomplete quadratic polynomial functions of the form [[$ f (x) = x^2 + bx $]] in the same coordinate system.
The graphs of the functions are congruent. The coefficient of the first-order term [[$ b $]] is clearly related to the second zero of the function and the position of the parabola's axis of symmetry.
- There are always two solutions to the equation.
- One of the solutions is always x = 0.
Equations of the form [[$ y = ax^2 + bx $]] are most conveniently solved by dividing them into factors.
- Divide the equation into the factors so that both factors contain the variable [[$ x $]].
- The zero-product property applies. The product is only zero if any of the factors is zero.
Example 1
Solve the equation [[$ 2x^2 + 6x = 0 $]] and sketch the graph of the parabola determined by the equation.
[[$ \begin{align*}
2x^2 + 6x &= 0 \;\;\; {\color {red} {\text {Divide the equation by factors.}}} \\
2x(x + 3) &= 0 \\
\end{align*} $]]
Apply the zero-product property.
[[$ \begin{align*}
2x &= 0 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; {\text {or}}\;\;\;\;\;\; \;\;\;\;\;\; x + 3 &= 0 \\
x &= 0 \;\;\;\;\;\; \;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\; x &= -3 \\
\end{align*} $]]
Answer: [[$ x = 0 $]] or [[$ x = -3 $]]
Example 2
Solve the equation [[$ 5x^2 - 2x = 0 $]] and sketch the graph of the parabola determined by the equation.
[[$ \begin{align*}
5x^2 - 2x &= 0 \;\;\; {\color {red} {\text {Divide the equation by factors.}}} \\
x(5x - 2) &= 0 \\
\end{align*} $]]
Apply the zero-product property.
[[$ \begin{align*} x &= 0 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; {\text {or}}\;\;\;\;\;\; \;\;\;\;\;\; 5x - 2 &= 0 \\ x &= 0 \;\;\;\;\;\; \;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\; 5x &= 2 \;\;\; {\color {blue} {||:5}} \\ x &&= \displaystyle\frac {2} {5} \end{align*} $]]Answer: [[$ x = 0 $]] or [[$ x = \displaystyle\frac {2} {5} $]]