Quadratic Formula

Front

What is the quadratic formula?

Back

The quadratic formula is given by x = (-b ± √(b² - 4ac)) / (2a), where a, b, and c are coefficients from the quadratic equation ax² + bx + c = 0.

Front

What does the discriminant (b² - 4ac) indicate in a quadratic equation?

Back

The discriminant indicates the nature of the roots of the quadratic equation: if it is positive, there are two distinct real roots; if it is zero, there is one real root (a repeated root); if it is negative, there are two complex roots.

Front

How do you identify the coefficients a, b, and c in the quadratic equation ax² + bx + c = 0?

Back

In the equation ax² + bx + c = 0, 'a' is the coefficient of x², 'b' is the coefficient of x, and 'c' is the constant term.

Front

What is the first step in solving a quadratic equation using the quadratic formula?

Back

The first step is to rearrange the equation into the standard form ax² + bx + c = 0, if it is not already in that form.

Front

How can you solve the equation 4x² + 4x + 1 = 0 using the quadratic formula?

Back

Identify a = 4, b = 4, c = 1. Calculate the discriminant: b² - 4ac = 4² - 4(4)(1) = 0. Since the discriminant is 0, there is one real root: x = -b/(2a) = -4/(2*4) = -½.

Front

What are the solutions to the equation m² - 5m - 14 = 0 using the quadratic formula?

Back

Identify a = 1, b = -5, c = -14. Calculate the discriminant: (-5)² - 4(1)(-14) = 25 + 56 = 81. The roots are x = (5 ± √81) / 2 = (5 ± 9) / 2, giving x = 7 and x = -2.

Front

How do you solve the equation 2x² - 36 = x using the quadratic formula?

Back

Rearrange to standard form: 2x² - x - 36 = 0. Identify a = 2, b = -1, c = -36. Calculate the discriminant: (-1)² - 4(2)(-36) = 1 + 288 = 289. The roots are x = (1 ± √289) / (2*2) = (1 ± 17) / 4, giving x = 9/2 and x = -4.

Front

What is the significance of the vertex in a quadratic function?

Back

The vertex of a quadratic function represents the maximum or minimum point of the parabola, depending on the direction it opens (upward or downward). It can be found using the formula x = -b/(2a).

Front

How can you determine the number of solutions to a quadratic equation graphically?

Back

Graph the quadratic function and observe the x-intercepts. The number of x-intercepts corresponds to the number of real solutions: no intercepts means no real solutions, one intercept means one real solution, and two intercepts mean two real solutions.

Front

What is the relationship between the roots of a quadratic equation and its graph?

Back

The roots of a quadratic equation are the x-values where the graph intersects the x-axis. The nature of the roots (real or complex) affects the graph's behavior.