3.5 Solving Quadratics by Completing the Square

Front

What is the process of completing the square?

Back

Completing the square is a method used to solve quadratic equations by rewriting the equation in the form (x - p)² = q, where p and q are constants.

Front

What is a perfect square trinomial?

Back

A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial, such as (x + a)² = x² + 2ax + a².

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 add it: x² + 6x + 9 = (x + 3)².

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

How do you solve the equation x² - 4x + 4 = 20 by completing the square?

Back

Rearrange to x² - 4x = 16, complete the square to get (x - 2)² = 16, then solve for x: x - 2 = ±4, so x = 6 or -2.

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.

Front

How do you create a perfect square trinomial from x² + 22x?

Back

Take half of 22 (which is 11), square it (11² = 121), and add it: x² + 22x + 121 = (x + 11)².

Front

What is the first step in solving r² - 4r = 30 by completing the square?

Back

Rearrange the equation to r² - 4r - 30 = 0, then move -30 to the other side: r² - 4r = 30.

Front

How do you find the roots of the equation x² + 12x = 5?

Back

Rearrange to x² + 12x - 5 = 0, complete the square to get (x + 6)² - 36 = 5, then solve for x.

Front

What is the formula for completing the square?

Back

The formula is: (x + b/2)² - (b/2)², where b is the coefficient of x in the quadratic expression.