- The constant to add to the binomial \(x^2 - 3x\) to make it a perfect square trinomial is \(c = (\frac{-3}{2})^2 = \frac{9}{4}\).
- The resulting perfect square trinomial is \(x^2 - 3x + \frac{9}{4}\).
- Factoring this trinomial yields \((x - \frac{3}{2})^2\).
- Final answer: Constant: \(\frac{9}{4}\), Trinomial: \(x^2 - 3x + \frac{9}{4}\), Factored: \((x - \frac{3}{2})^2\).
Explanation
- Finding the Constant and the Trinomial
We are given the binomial \(x^2 - 3x\). We want to add a constant to this expression such that the result is a perfect square trinomial. A perfect square trinomial of the form \(x^2 + bx + c\) is created when \(c = (\frac{b}{2})^2\). Here, the coefficient of the linear term \(x\) is \(b = -3\). To complete the square, we calculate \(c = (\frac{-3}{2})^2 = \frac{9}{4}\). Adding this constant to the expression yields the perfect square trinomial \(x^2 - 3x + \frac{9}{4}\).
- Factoring the Trinomial
Now we factor the trinomial \(x^2 - 3x + \frac{9}{4}\). A perfect square trinomial \(x^2 + bx + (\frac{b}{2})^2\) factors into \((x + \frac{b}{2})^2\). Substituting \(b = -3\), we get \((x - \frac{3}{2})^2\). Thus, the trinomial \(x^2 - 3x + \frac{9}{4}\) is equivalent to \((x - \frac{3}{2})^2\).
Examples
Completing the square is a powerful tool in algebra used for solving quadratic equations, deriving the quadratic formula, and graphing parabolas. For instance, if you are modeling the path of a projectile (like a thrown ball), you often encounter quadratic functions. By completing the square, you can easily find the vertex of the parabola, which represents the maximum height reached by the ball. This helps in understanding how long an object stays in the air and where it will land.