Completing the Square

Front

What is 'completing the square'?

Back

Completing the square is a method used to solve quadratic equations by rewriting them in the form (x - p)² = q, making it easier to find the roots.

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

How do you complete the square for the expression x² + 6x?

Back

To complete the square, take half of the coefficient of x (which is 6), square it (3² = 9), and rewrite the expression as (x + 3)² - 9.

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 first step in completing the square for the equation x² - 4x = 20?

Back

The first step is to move the constant to the other side: x² - 4x - 20 = 0.

Front

What do you add to both sides of the equation x² - 4x = 20 to complete the square?

Back

You add (4/2)² = 4 to both sides, resulting in (x - 2)² = 24.

Front

How do you find the roots of the equation (x - 2)² = 24?

Back

Take the square root of both sides: x - 2 = ±√24, then solve for x to get x = 2 ± 2√6.

Front

What is the significance of the term 'b² - 4ac' in a quadratic equation?

Back

The term 'b² - 4ac' is called the discriminant and determines the nature of the roots: if positive, two real roots; if zero, one real root; if negative, no real roots.

Front

How do you solve the equation y² + 10y = -9 by completing the square?

Back

Rearrange to y² + 10y + 25 = 16, then factor to (y + 5)² = 16, leading to y = -5 ± 4.

Front

What is the completed square form of x² + 6x + 9?

Back

The completed square form is (x + 3)².