Solving quadratic Equations by completing the Squares

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 of C when completing the square for the expression x² + bx + C?

Back

To find C, take half of the coefficient of x (which is b), square it, and add it to the expression. C = (b/2)².

Front

What are the roots of a quadratic equation?

Back

The roots of a quadratic equation are the values of x that satisfy the equation, which can be found using methods such as factoring, completing the square, or the quadratic formula.

Front

What is the quadratic formula?

Back

The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a), used to find the roots of a quadratic equation.

Front

How do you solve the equation n² - 2n - 3 = 0 by completing the square?

Back

To solve by completing the square, rewrite the equation as (n - 1)² - 4 = 0, then solve for n to find n = 3 and n = -1.

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, two complex roots.

Front

How can you express a quadratic equation in vertex form?

Back

A quadratic equation can be expressed in vertex form as y = a(x - h)² + k, where (h, k) is the vertex of the parabola.

Front

What does it mean for a quadratic equation to have complex roots?

Back

Complex roots occur when the discriminant is negative, resulting in roots that are not real numbers and can be expressed in the form a ± bi.

Front

How do you solve 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, giving roots y = -1 and y = -9.