Find all real solutions of the equation by completing the square \( x^{2}-12 x+32=21 \). \( x_{1}=\square \) with \( x_{1}

Ask by Watkins Rodriguez. in the United States

Mar 09,2025

To solve the equation \( x^{2}-12x+32=21 \) by completing the square, first rearrange it as \( x^{2}-12x+11=0 \). Next, complete the square on the left side: 1. Take half of the coefficient of \( x \) (which is -12), square it to get \( 36 \), and rewrite the equation: \[ (x-6)^{2} - 36 + 11 = 0 \Rightarrow (x-6)^{2} = 25. \] 2. Solve for \( x \): \[ x-6 = 5 \quad \text{or} \quad x-6 = -5. \] 3. Thus, the solutions are \( x_1=1 \) and \( x_2=11 \). So, \( x_{1} = 1 \) with \( x_{1} < x_{2} \). The completed square method not only gives you the roots, but also reveals how quadratic equations can model various phenomena. For instance, the vertex form of a parabola \( y = a(x-h)^2 + k \) allows you to easily find the maximum or minimum values, making it handy in optimization problems ranging from business to engineering! If you want to delve deeper into this concept, consider exploring texts on quadratic functions and their properties. Books like "Precalculus" by Michael Sullivan or "Algebra and Trigonometry" by Robert F. Blitzer not only cover completing the square in depth but also provide engaging problems and real-life applications. Happy studying!