Quadratic Formula

Front

What is the quadratic formula?

Back

The quadratic formula is given by x = (-b ± √(b² - 4ac)) / (2a), used to find the roots of a quadratic equation ax² + bx + c = 0.

Front

What does the discriminant indicate in a quadratic equation?

Back

The discriminant, calculated as b² - 4ac, indicates the nature of the roots of the quadratic equation. If it is positive, there are two distinct real roots; if zero, there is one real root; if negative, there are two complex roots.

Front

How do you identify the coefficients a, b, and c in a quadratic equation?

Back

In the standard form of a quadratic 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 significance of the leading coefficient in a quadratic equation?

Back

The leading coefficient 'a' determines the direction of the parabola (upward if a > 0, downward if a < 0) and affects the width of the parabola.

Front

How can you determine the vertex of a parabola represented by a quadratic equation?

Back

The vertex of the parabola can be found using the formula (-b/(2a), f(-b/(2a))) where f(x) is the quadratic function.

Front

What is the axis of symmetry in a quadratic function?

Back

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves, given by the equation x = -b/(2a).

Front

What are the steps to solve a quadratic equation using the quadratic formula?

Back

1. Identify coefficients a, b, and c. 2. Calculate the discriminant (b² - 4ac). 3. Substitute a, b, and the discriminant into the quadratic formula. 4. Simplify to find the roots.

Front

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

Back

The roots of a quadratic equation ax² + bx + c = 0 correspond to the values of x that make the equation equal to zero, and can be expressed as factors: a(x - r1)(x - r2) where r1 and r2 are the roots.

Front

How do you convert a quadratic equation from standard form to vertex form?

Back

To convert from standard form y = ax² + bx + c to vertex form y = a(x - h)² + k, complete the square to find the vertex (h, k).

Front

What is the significance of the y-intercept in a quadratic function?

Back

The y-intercept is the point where the graph intersects the y-axis, found by evaluating the function at x = 0, which corresponds to the constant term c in the equation.