Algebra Class: Completing the Square

What is the value of c in the perfect square trinomial x² + 10x + c?

What number would you add to both sides of x² - 20x = 5 to complete the square?

What number would you add to both sides of x² + 7x = 4 to complete the square?

In the equation x² + 12x = -20, what value should be added to both sides to complete the square?

What is the solution set of the equation x² + 8x = 33 when solved by completing the square?

What describes the solutions of the equation p² - 14p - 72 = 0?

What is the solution set of x² - 10 = 30x?

What are the solutions of the equation x² = 8 - 5x?

Is Adiya's method of solving the equation x² + 6 = 20x correct?

Completing the Square: Key Concepts

Completing the square involves transforming a quadratic equation into a perfect square trinomial to simplify solving for roots.
A perfect square trinomial is of the form (x^2 + 2bx + b^2).

Example Problems and Solutions

For (x^2 + 10x + c), the value of (c) required to complete the square is 25, resulting in ((x + 5)^2).
To complete the square for (x^2 - 20x = 5), add 100 to both sides, transforming the left side into ((x - 10)^2).
Completing the square on (x^2 + 7x = 4) involves adding 12.25 (the square of 3.5) to both sides.

Solving Quadratic Equations

For the equation (x^2 + 12x = -20), add 36 to both sides. This leads to ((x + 6)^2 = 16), yielding (x = -2) or (x = -10).
In (x^2 + 8x = 33), the completed square provides a solution set represented as A (exact solutions not specified).

Analyzing Equations

The equation (p^2 - 14p - 72 = 0) can be solved by completing the square, where adding 49 results in a binomial square on the left side and 121 on the right. The solutions are denoted as A.
The solution set for (x^2 - 10 = 30x) and (x^2 = 8 - 5x) are given as C and B, respectively.

Common Misconceptions

Adiya's method for solving (x^2 + 6 = 20x) by dividing the linear coefficient and squaring it is incorrect. The constant must be isolated first, followed by squaring half of the linear coefficient for perfect square formation. The correct method involves rearranging terms before applying completing the square.

Test your knowledge of solving quadratic equations by completing the square. This quiz includes flashcards that cover key concepts and values needed for mastering the technique. Challenge yourself to find the correct values to complete the square for various expressions.