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 or (a - b)² = a² - 2ab + b².

Front

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

Back

To find c, take half of the coefficient of x (which is -14), square it: (-14/2)² = 49. Thus, c = 49.

Front

What is the vertex form of a quadratic function?

Back

The vertex form of a quadratic function is given by y = a(x - h)² + k, where (h, k) is the vertex of the parabola.

Front

How do you complete the square for the equation x² + 12x = 5?

Back

1. Move 5 to the other side: x² + 12x - 5 = 0. 2. Take half of 12, square it: (12/2)² = 36. 3. Add 36 to both sides: x² + 12x + 36 = 41. 4. Factor: (x + 6)² = 41.

Front

What are the solutions to the equation x² + 12x - 5 = 0 after completing the square?

Back

The solutions are x = -6 ± √41.

Front

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

Back

The vertex (h, k) can be found using h = -b/(2a) and k = f(h).

Front

Back

Front

What is the equivalent expression for the perfect square trinomial x² - 18x + 81?

Back

The equivalent expression is (x - 9)².

Front

How do you find the vertex of the quadratic function y = 4x² + 24x + 5?

Back

Use h = -b/(2a): h = -24/(2*4) = -3. Then find k by substituting h into the function: k = 4(-3)² + 24(-3) + 5 = -31. Vertex is (-3, -31).

Front

What is the discriminant of a quadratic equation and what does it indicate?

Back

The discriminant is given by D = b² - 4ac. It indicates the nature of the roots: D > 0 means two real roots, D = 0 means one real root, D < 0 means two complex roots.