- For \(x^2 + bx + c\), use the property \(c = (\frac{b}{2})^2\) and \(b = 2\sqrt{c}\).
- For \(x^2 - 10x + \boxed{25}\), take \(\frac{-10}{2} = -5\) and square it to get \(25\).
- For \(x^2 + \boxed{12}x + 36\), take \(\sqrt{36} = 6\) and multiply by 2 to get \(12\).
- For \(x^2 + \frac{1}{2}x + \boxed{\frac{1}{16}}\), take \(\frac{1}{2} \div 2 = \frac{1}{4}\) and square it to get \(\frac{1}{16}\).
- Final answers: \(\boxed{25}, \boxed{12}, \boxed{\frac{1}{16}}\).
Explanation
- Understanding the Perfect-Square Trinomial Property
To form a perfect-square trinomial of the form \(x^2 + bx + c\), we need to satisfy the condition \((x + k)^2 = x^2 + 2kx + k^2\). By comparing the terms, we see that \(b = 2k\) and \(c = k^2\), which implies \(k = \frac{b}{2}\) and \(c = (\frac{b}{2})^2\). This means the constant term \(c\) must be the square of half the coefficient of \(x\). If the constant term is given, we find \(b\) by calculating \(\pm 2\sqrt{c}\).
- Solving for the First Expression
For the expression \(x^2 - 10x + \boxed{\phantom{00}}\), we identify \(b = -10\). To make this a perfect-square trinomial, the missing constant term is:
\[\left(\frac{b}{2}\right)^2 = \left(\frac{-10}{2}\right)^2 = (-5)^2 = 25.\]
Thus, the expression is \(x^2 - 10x + 25\), which factors as \((x - 5)^2\).
- Solving for the Second Expression
For the expression \(x^2 + \boxed{\phantom{00}}x + 36\), we have the constant term \(c = 36\). Since \(c = k^2\), we have \(k = \pm\sqrt{36} = \pm 6\). The linear coefficient \(b\) is then \(2k = 2(\pm 6) = \pm 12\). Choosing the standard positive value, the coefficient is \(12\), making the expression \(x^2 + 12x + 36\), which factors as \((x + 6)^2\).
- Solving for the Third Expression
For the expression \(x^2 + \frac{1}{2}x + \boxed{\phantom{00}}\), we have \(b = \frac{1}{2}\). To make this a perfect-square trinomial, the missing constant term is:
\[\left(\frac{b}{2}\right)^2 = \left(\frac{1/2}{2}\right)^2 = \left(\frac{1}{4}\right)^2 = \frac{1}{16}.\]
Thus, the expression is \(x^2 + \frac{1}{2}x + \frac{1}{16}\), which factors as \((x + \frac{1}{4})^2\).
- Conclusion
The values required to complete the perfect-square trinomials are:
\(x^2 - 10x + \boxed{25}\)
\(x^2 + \boxed{12}x + 36\)
\(x^2 + \frac{1}{2}x + \boxed{\frac{1}{16}}\)
All calculations follow the property \(c = (\frac{b}{2})^2\) and \(b = 2\sqrt{c}\).
Examples
Perfect-square trinomials are extremely useful in geometry and physics, especially when completing the square to find the vertex of a parabola or the center and radius of a circle from its standard equation. For example, if you are modeling the trajectory of a ball, the equation might be given in a form that requires completing the square to identify the peak height (vertex) of the flight. Mastery of these algebraic completions allows you to transition between different representations of quadratic functions to better interpret physical behavior.