Solving quadratic Equations by completing the Squares

Front

What is the standard form of a quadratic equation?

Back

The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

Front

What does it mean to complete the square in a quadratic equation?

Back

Completing the square involves rewriting a quadratic equation in the form (x - p)² = q, which makes it easier to solve for x.

Front

How do you complete the square for the equation x² + 6x = 5?

Back

To complete the square, add (6/2)² = 9 to both sides, resulting in (x + 3)² = 14.

Front

What is the first step in solving a quadratic equation by completing the square?

Back

The first step is to move the constant term to the other side of the equation.

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

What is the relationship between the solutions of a quadratic equation and its graph?

Back

The solutions of a quadratic equation correspond to the x-intercepts of its graph.

Front

How can you determine the number of real solutions a quadratic equation has?

Back

You can determine the number of real solutions by calculating the discriminant (b² - 4ac). If it's positive, there are two solutions; if zero, one solution; if negative, no real solutions.

Front

What is the discriminant of the quadratic equation x² - 4x + 4 = 0?

Back

The discriminant is (−4)² - 4(1)(4) = 0, indicating one real solution.

Front

How do you find the solutions of the equation (x + 4)² = 7?

Back

Take the square root of both sides: x + 4 = ±√7, then solve for x to get x = -4 ± √7.

Front

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

Back

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