4.5 Completing the Square

Front

What is a perfect square trinomial?

Back

A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial, typically in the form (a + b)² = a² + 2ab + b².

Front

How do you find the value of c to complete the square for x² + 2x + c?

Back

To find c, take half of the coefficient of x (which is 2), square it (1), and add it to the expression. Thus, c = 1.

Front

What is the first step in completing the square for the equation x² + 4x + 1 = 0?

Back

The first step is to move the constant term to the other side: x² + 4x = -1.

Front

What is the formula for completing the square?

Back

The formula is: x² + bx = (x + b/2)² - (b/2)².

Front

How do you solve x² + 4x + 1 = 0 by completing the square?

Back

1. Move 1 to the other side: x² + 4x = -1. 2. Complete the square: (x + 2)² = 3. 3. Solve for x: x = -2 ± √3.

Front

What is the solution to (2x - 1)² = 49?

Back

To solve, take the square root of both sides: 2x - 1 = ±7. Thus, x = 4 or x = -3.

Front

How do you find the missing value c in x² + 26x + c to make it a perfect square?

Back

Take half of 26 (which is 13), square it to get 169. So, c = 169.

Front

What is the relationship between the coefficients in a perfect square trinomial?

Back

In a perfect square trinomial of the form a² + 2ab + b², the coefficient of x (2ab) is twice the product of a and b.

Front

What does it mean to factor a quadratic expression?

Back

Factoring a quadratic expression means rewriting it as a product of its linear factors, typically in the form (x - p)(x - q).

Front

What is the significance of the vertex in completing the square?

Back

The vertex form of a quadratic equation, y = a(x - h)² + k, shows the vertex (h, k) of the parabola, which is useful for graphing.