Practice Question 5.2 \( \begin{array}{l}\text { (a) Express } 3 x^{2}+x-3 \text { in the form } a(x-h)^{2}+k \text {, where } a, h \text { and } k \text { are constants. } \\ \text { since positive } 3\left(x^{2}+\right.\end{array} \)
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To express the quadratic \(3x^2 + x - 3\) in the form \(a(x-h)^2 + k\), we start with completing the square. First, factor out the 3 from the \(x\) terms: \[ 3\left(x^2 + \frac{1}{3}x\right) - 3. \] Now, to complete the square inside the parentheses, take half of \(\frac{1}{3}\), which is \(\frac{1}{6}\), square it to get \(\frac{1}{36}\), and add/subtract this inside the square: \[ 3\left(x^2 + \frac{1}{3}x + \frac{1}{36} - \frac{1}{36}\right) - 3. \] This simplifies to: \[ 3\left(\left(x + \frac{1}{6}\right)^2 - \frac{1}{36}\right) - 3, \] which becomes: \[ 3\left(x + \frac{1}{6}\right)^2 - \frac{1}{12} - 3. \] Combining the constants gives: \[ 3\left(x + \frac{1}{6}\right)^2 - \frac{37}{12}. \] Thus, the constants are \(a = 3\), \(h = -\frac{1}{6}\), and \(k = -\frac{37}{12}\). For practice and deeper understanding, exploring transformations of quadratic functions can be beneficial. Try graphing \(3x^2 + x - 3\) and its vertex form. Understanding how the vertex changes with \(h\) and \(k\) is crucial for mastering quadratics! Remember that when completing the square, one common mistake is forgetting to properly balance the equation after adding the constant. Always ensure you add and subtract the same number to maintain equality! This tip is vital to avoid errors during transformation into vertex form.
