3.5 Solving Quadratics by Completing the Square

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 (x - p)² = q, where p and q are constants. This method allows for easier solving of the equation.

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 find the value to complete the square for the expression x² - 4x?

Back

To complete the square, take half of the coefficient of x (which is -4), square it, and add it: (-4/2)² = 4. Thus, x² - 4x + 4 = (x - 2)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

Solve the equation by completing the square: x² + 6x - 4 = 36. What are the roots?

Back

x = 4, -10.

Front

Complete the square for the expression x² + 6x. What is the constant term needed?

Back

The constant term needed is 9, so x² + 6x + 9 = (x + 3)².

Front

What is the first step in completing the square for the equation x² + 6x = 5?

Back

Move the constant to the other side: x² + 6x = 5.

Front

What is the solution to the equation r² - 4r = 30 after completing the square?

Back

The solutions are r = 2 ± √34.

Front

How do you rewrite the equation x² + 6x + ____ = 0 by completing the square?

Back

Add 9 to both sides to get (x + 3)² = 9.

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.