Complete the Square Flashcards

Which of the following expressions are perfect-square trinomials? (Select all that apply)

Complete the expression so it forms a perfect-square trinomial: x² - 5x + ____

A student says that if 5x² = 20, then x must be equal to 2. Do you agree or disagree with the student? Justify your answer.

Given (x - 7)² = 36, select the values of x.

Given (x - 1)² = 50, select the values of x.

What method would you choose to solve the equation 2x² - 7 = 9? Explain why you chose this method.

Solve x² - 16x + ____ = -12 by completing the steps. First, subtract from each side of the equation.

Next, add ____ to each side of the equation to complete the square.

Now, write x² - 16x + 64 = -8 as ____.

Take the square root of both sides to get the solutions.

Complete the square: x² - 6x + ____ = -13 + ____

Factor the trinomial and simplify: (x + ____ )² = ____.

The solution to x² - 10x = 24 is ____.

The solution to 2x² - 11 = 87 is ____.

The solution to 3x² - 12x + 24 = 0 is ____.

Complete the square to write 16t² - 96t + 48 = 0 as ____.

Solve (t - 3)² = 6. The arrow is at a height of 48 ft after approximately ____.

Perfect-Square Trinomials

Perfect-square trinomials include expressions that can be factored into the form (a + b)².
From the examples provided, the following are perfect-square trinomials:
- 4x² + 12x + 9
- x² + 20x + 100
- x² + 4x + 16

Completing the Square

Completing the square involves transforming a quadratic expression into a perfect-square trinomial.
For x² - 5x, adding (25/4) completes the square.

Equation Solutions

If 5x² = 20, the solutions are x = 2 and x = -2, showcasing the principle of finding values for x in quadratic equations.
Given (x - 7)² = 36 leads to solutions x = 13 and x = 1 through taking the square root of both sides.
For (x - 1)² = 50, the valid solutions are x = 51, reaffirming the value of checking multiple potential solutions.

Equation Solving Processes

To solve the equation 2x² - 7 = 9:
- Divide all terms by two, simplifying the equation.
- After rearranging, the solution involves completing the square.

Completing Steps in Quadratics

To solve x² - 16x = -12 by completing the square:
- First, determine the value needed to balance the equation, which is 60.
- Then, add 64 to both sides, allowing the equation to be expressed in a perfect-square form.

Final Steps in Solutions

For the expression x² - 16x + 64 = -8, solving helps identify roots corresponding to the square root of both sides, yielding solutions of x = 8 and x = 2.

Factoring and Simplifying

The trinomial can be factored and simplified to the form (x + b)², where specific values like -3 and -4 play a crucial role.

Quadratic Solutions

Solutions for various quadratic equations include:
- For x² - 10x = 24, the result is indicated as solution C.
- For 2x² - 11 = 87, the solution is acknowledged as A.
- The solution to 3x² - 12x + 24 = 0 is denoted as B.

Contextual Application

Completing the square can also transform equations such as 16t² - 96t + 48 = 0 into simpler forms, helping clearly understand the roots.
Solving (t - 3)² = 6 yields height problems leading to solutions around 5 and 5.45 for applications like projectile motion.

Test your understanding of perfect-square trinomials and completing the square with these flashcards. This quiz covers various expressions and their properties, encouraging you to engage with key concepts in quadratic equations.