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 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 equation x² - 4x = 0?

Back

To complete the square, take half of the coefficient of x (which is -4), square it (which gives 4), and add it to both sides: (x - 2)² = 4.

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 find the roots of a quadratic equation using completing the square?

Back

To find the roots, rewrite the equation in the form (x - p)² = q, then take the square root of both sides and solve for x.

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, there are two distinct real roots; if zero, one real root; if negative, no real roots.

Front

Solve the equation by completing the square: x² + 8x + 12 = 0.

Back

x = -4 ± √4, which gives x = -2 and x = -6.

Front

What is the first step in completing the square for the equation x² + 6x = 8?

Back

Move the constant to the other side: x² + 6x - 8 = 0.

Front

Complete the square for the expression x² + 10x.

Back

x² + 10x = (x + 5)² - 25.

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.