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 + 10x?

Back

To complete the square, take half of the coefficient of x (which is 10), square it (25), and rewrite the expression as (x + 5)² - 25.

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 x² + 6x - 4 = 36?

Back

First, move the constant to the other side: x² + 6x = 40.

Front

What do you add to both sides when completing the square for x² + 6x?

Back

Add (6/2)² = 9 to both sides to complete the square.

Front

What is the completed square form of x² + 6x - 4 = 36?

Back

The completed square form is (x + 3)² = 49.

Front

How do you find the roots after completing the square?

Back

Take the square root of both sides and solve for x: x + 3 = ±7, leading to x = 4 and x = -10.

Front

What is the solution to (2x - 1)² = 49?

Back

The solutions are x = 7 and x = -3.

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, no real roots.