Example 1
Completion requirements
| Example 1 |
Graphically solve the equation \(0.75x^2 = -2x + 7\), using technology.
Step 1:
Rearrange the equation so one side is equal to zero.
\(\begin{align}
0.75x^2 &= -2x + 7 \\
0.75x^2 + 2x - 7 &= -2x + 2x + 7 - 7 \\
0.75x^2 + 2x - 7 &= 0 \\
\end{align}\)
Step 2: Graph the equation of the corresponding function, \(y = 0.75x^2 + 2x - 7\), using technology.
Step 3: Find the \(x\)-intercepts of the graph, which correspond to the roots of the quadratic equation.
The \(x\)-intercepts of the graph, and thus the roots of the equation, are \(2\) and approximately \(–4.67\).
Step 4: Verify the roots.
Solving equations using technology often gives approximate (rounded) solutions. The value of \(–4.67\) is an approximate solution. As such, the left side and the right side of the verification are not quite equal.
\(\begin{align}
0.75x^2 &= -2x + 7 \\
0.75x^2 + 2x - 7 &= -2x + 2x + 7 - 7 \\
0.75x^2 + 2x - 7 &= 0 \\
\end{align}\)
Step 2: Graph the equation of the corresponding function, \(y = 0.75x^2 + 2x - 7\), using technology.
Step 3: Find the \(x\)-intercepts of the graph, which correspond to the roots of the quadratic equation.
The \(x\)-intercepts of the graph, and thus the roots of the equation, are \(2\) and approximately \(–4.67\).
Step 4: Verify the roots.
| Left Side |
Right Side
|
|---|---|
|
\(\begin{array}{r}
0.75x^2 \\ 0.75\left( { -4.67} \right)^2 \\ 16.356675 \\ \end{array}\) |
\(\begin{array}{l}
-2x + 7 \\ -2\left( { -4.67} \right)+ 7 \\ 16.34 \\ \end{array}\) |
| LS = RS | |
| Left Side |
Right Side
|
|---|---|
| \(\begin{array}{r} 0.75x^2 \\ 0.75\left( 2 \right)^2 \\ 3 \\ \end{array}\) |
\(\begin{array}{l} -2x + 7 \\ -2\left( 2 \right) + 7 \\ 3 \\ \end{array}\) |
| LS = RS | |
Solving equations using technology often gives approximate (rounded) solutions. The value of \(–4.67\) is an approximate solution. As such, the left side and the right side of the verification are not quite equal.