5. 6 Complete the Square Assignment

Front

What is the process of completing the square in a quadratic equation?

Back

Completing the square involves rewriting a quadratic equation in the form of (x + p)^2 = q, which allows for easier solving and understanding of the equation's properties.

Front

What is the standard form of a quadratic equation?

Back

The standard form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.

Front

How do you complete the square for the equation x^2 + 12x = 5?

Back

1. Move 5 to the other side: x^2 + 12x - 5 = 0. 2. Take half of 12 (which is 6), square it (36), and add to both sides: x^2 + 12x + 36 = 41. 3. Factor: (x + 6)^2 = 41. 4. Solve: x + 6 = ±√41, so x = -6 ± √41.

Front

What is the solution to the equation x^2 + 6x + 27 = 0 using completing the square?

Back

1. Move 27 to the other side: x^2 + 6x = -27. 2. Take half of 6 (which is 3), square it (9), and add to both sides: x^2 + 6x + 9 = -27 + 9. 3. Factor: (x + 3)^2 = -18. 4. Solve: x + 3 = ±√(-18), so x = -3 ± 3i√2.

Front

What does it mean if a quadratic equation has no real solutions?

Back

It means that the discriminant (b^2 - 4ac) is less than zero, indicating that the parabola does not intersect the x-axis.

Front

What is the discriminant of a quadratic equation?

Back

The discriminant is the expression b^2 - 4ac in the quadratic formula, which determines the nature of the roots of the equation.

Front

How do you find the vertex of a parabola given by a quadratic equation?

Back

The vertex can be found using the formula (-b/2a, f(-b/2a)), where f(x) is the quadratic function.

Front

What is the vertex form of a quadratic equation?

Back

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

Front

Back

Front

What is the significance of the coefficient 'a' in a quadratic equation?

Back

The coefficient 'a' determines the direction of the parabola (upward if a > 0, downward if a < 0) and affects the width of the parabola.