Solving Quadratics with Real and Imaginary Solutions

Front

What is a quadratic equation?

Back

A quadratic equation is a polynomial equation of degree 2, typically in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

Front

What is the quadratic formula?

Back

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

Front

What does the discriminant of a quadratic equation indicate?

Back

The discriminant, given by b² - 4ac, indicates the nature of the roots: if > 0, two distinct real roots; if = 0, one real root; if < 0, two complex (imaginary) roots.

Front

Solve x² - 5x + 10 = 0 using the quadratic formula.

Back

x = (5 - i√15)/2, (5 + i√15)/2.

Front

Solve 2x² - 9x - 35 = 0 using the quadratic formula.

Back

x = -5/2, 7.

Front

What are real solutions in the context of quadratic equations?

Back

Real solutions are the values of x that satisfy the quadratic equation and are found on the real number line.

Front

What are imaginary solutions in the context of quadratic equations?

Back

Imaginary solutions occur when the discriminant is negative, resulting in complex numbers that include the imaginary unit i.

Front

Solve x² + 4x - 40 = -8 using the quadratic formula.

Back

x = -8, 4.

Front

Solve 3x² - 6x + 6 = 0 using the quadratic formula.

Back

x = 1 + i, 1 - i.

Front

What is the vertex form of a quadratic equation?

Back

The vertex form is y = a(x - h)² + k, where (h, k) is the vertex of the parabola.