Verified Answer :
- Identify the form of a perfect-square trinomial: \(x^2 \pm 2ax + a^2 = (x \pm a)^2\).
- For the first expression \(x^2 - 5x + \boxed{\phantom{0}}\), calculate the constant term by halving the coefficient of \(x\) and squaring it: \(\left(\frac{-5}{2}\right)^2 = 6.25\).
- For the second expression \(x^2 + \boxed{\phantom{0}} \times x + 49\), find the linear coefficient by identifying the square root of the constant term: \(\pm 2 \cdot \sqrt{49} = \pm 14\).
- Final results are: 1. \(\boxed{6.25}\), 2. \(\boxed{\pm 14}\).
Explanation
- Understanding Perfect-Square Trinomials
To form a perfect-square trinomial, an expression must match the form of the square of a binomial: \((x + a)^2 = x^2 + 2ax + a^2\) or \((x - a)^2 = x^2 - 2ax + a^2\). Our objective is to find the missing terms that satisfy these conditions for the given expressions.
- Completing the First Expression
Consider the expression \(x^2 - 5x + c\). To make this a perfect square, it must match the form \(x^2 - 2ax + a^2 = (x - a)^2\). By comparing the linear term \(-5x\) with \(-2ax\), we set \(-2a = -5\), which gives \(a = \frac{5}{2} = 2.5\). The constant term must then be \(a^2 = \left(\frac{5}{2}\right)^2 = \frac{25}{4} = 6.25\). Thus, the complete trinomial is \(x^2 - 5x + 6.25\).
- Completing the Second Expression
Consider the expression \(x^2 + b \cdot x + 49\). To make this a perfect square, it must match the form \((x + a)^2 = x^2 + 2ax + a^2\). Comparing the constant term \(49\) with \(a^2\), we have \(a^2 = 49\), which implies \(a = 7\) or \(a = -7\). The linear coefficient is \(2a\). If \(a = 7\), the coefficient is \(2(7) = 14\). If \(a = -7\), the coefficient is \(2(-7) = -14\). Thus, the expression can be completed as \(x^2 + 14x + 49\) or \(x^2 - 14x + 49\).
Examples
Completing the square is a fundamental skill in algebra with many practical applications. For example, if you are designing a parabolic reflector or calculating the trajectory of a projectile, the equation of the path is often expressed as a quadratic function. Converting this function into vertex form, which relies on completing the square, allows you to easily identify the highest or lowest point of the path (the vertex), which is crucial for maximizing efficiency or predicting where an object will land.