Math Problem Statement

2(x+2)(x-3), expand and complete the square

Solution

Let's first expand the expression 2(x+2)(x3)2(x+2)(x-3).

Step 1: Expand the expression

We can start by expanding (x+2)(x3)(x+2)(x-3) using the distributive property (FOIL method):

(x+2)(x3)=x23x+2x6=x2x6(x+2)(x-3) = x^2 - 3x + 2x - 6 = x^2 - x - 6

Now, multiply this result by 2:

2(x2x6)=2x22x122(x^2 - x - 6) = 2x^2 - 2x - 12

So, the expanded expression is:

2x22x122x^2 - 2x - 12

Step 2: Complete the square

To complete the square for the quadratic expression 2x22x122x^2 - 2x - 12, we need to follow these steps:

  1. Factor out the coefficient of x2x^2 (which is 2) from the first two terms:

2(x2x)122(x^2 - x) - 12

  1. Complete the square inside the parentheses. To do this, take the coefficient of xx (which is -1), divide it by 2, and square it:

(12)2=14\left(\frac{-1}{2}\right)^2 = \frac{1}{4}

So, we add and subtract 14\frac{1}{4} inside the parentheses:

2(x2x+1414)122\left(x^2 - x + \frac{1}{4} - \frac{1}{4}\right) - 12

  1. Simplify the expression inside the parentheses:

2((x12)214)122\left( \left(x - \frac{1}{2}\right)^2 - \frac{1}{4} \right) - 12

  1. Distribute the 2:

2(x12)22×14122\left(x - \frac{1}{2}\right)^2 - 2 \times \frac{1}{4} - 12

2(x12)212122\left(x - \frac{1}{2}\right)^2 - \frac{1}{2} - 12

  1. Simplify the constants:

2(x12)21212=2(x12)22522\left(x - \frac{1}{2}\right)^2 - \frac{1}{2} - 12 = 2\left(x - \frac{1}{2}\right)^2 - \frac{25}{2}

So, the expression 2x22x122x^2 - 2x - 12 in completed square form is:

2(x12)22522\left(x - \frac{1}{2}\right)^2 - \frac{25}{2}


Would you like more details or have any questions on this?

Here are some related questions you might find interesting:

  1. How do you complete the square for expressions that aren't factored?
  2. What is the vertex form of a quadratic equation?
  3. Can completing the square help in solving quadratic equations? If so, how?
  4. How does completing the square relate to the quadratic formula?
  5. What is the significance of the completed square form in graphing parabolas?

Tip: When completing the square, always remember to add and subtract the same value inside the parentheses to keep the expression equivalent.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Expressions
Expanding Expressions
Completing the Square

Formulas

(a+b)(a+c) = a^2 + ac + bc + bc
Completing the square method: (x + b/2)^2 - (b/2)^2

Theorems

Distributive Property (FOIL method)
Completing the Square Theorem

Suitable Grade Level

Grades 9-11