5.5&5.6 HW Completing the Square and Solving Complex

Front

What is a perfect square trinomial?

Back

A perfect square trinomial is a quadratic expression that can be expressed as the square of a binomial. It takes the form: (a + b)² = a² + 2ab + b².

Front

How do you find the value of c that makes x² - 14x + c a perfect square trinomial?

Back

To find c, use the formula c = (b/2)², where b is the coefficient of x. Here, b = -14, so c = (-14/2)² = 49.

Front

What is the vertex of a quadratic function in the form y = ax² + bx + c?

Back

The vertex can be found using the formula: x = -b/(2a). For y = 4x² + 24x + 5, the vertex is at (-3, -31).

Front

How do you simplify the square root of a number, such as \sqrt{200}?

Back

To simplify \sqrt{200}, factor it into its prime factors: \sqrt{200} = \sqrt{100 \cdot 2} = \sqrt{100} \cdot \sqrt{2} = 10\sqrt{2}.

Front

What is the factored form of the expression x² - 4x + 4?

Back

The expression can be factored as (x - 2)².

Front

What is the relationship between the coefficients of a quadratic equation and its vertex?

Back

The vertex (h, k) of a quadratic equation y = ax² + bx + c can be found using h = -b/(2a) and k = f(h).

Front

What does it mean to complete the square?

Back

Completing the square is a method used to convert a quadratic equation into a perfect square trinomial, making it easier to solve or graph.

Front

How do you convert the equation x² - 4x + 4 = 20 into vertex form?

Back

First, move 20 to the left: x² - 4x - 20 + 4 = 0. Then, complete the square: (x - 2)² = 20.

Front

What is the significance of the discriminant in a quadratic equation?

Back

The discriminant (b² - 4ac) determines the nature of the roots: if positive, two real roots; if zero, one real root; if negative, two complex roots.

Front

How do you identify a perfect square trinomial from an expression?

Back

A perfect square trinomial can be identified if it can be expressed as (x - a)² or (x + a)², where the middle term is twice the product of the square roots of the first and last terms.