A2 Quadratic Formula and Discriminant

Front

What is the quadratic formula?

Back

The quadratic formula is given by x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where a, b, and c are coefficients of the quadratic equation ax^2 + bx + c = 0.

Front

What does the discriminant of a quadratic equation indicate?

Back

The discriminant, given by D = b^2 - 4ac, indicates the nature of the roots of the quadratic equation. If D > 0, there are two distinct real solutions; if D = 0, there is one real solution (a repeated root); if D < 0, there are two imaginary solutions.

Front

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

Back

In a quadratic equation of the form ax^2 + bx + c = 0, 'a' is the coefficient of x^2, 'b' is the coefficient of x, and 'c' is the constant term.

Front

What is the significance of a negative discriminant?

Back

A negative discriminant indicates that the quadratic equation has two imaginary solutions, meaning the graph of the equation does not intersect the x-axis.

Front

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

Back

To convert from standard form (ax^2 + bx + c) to vertex form (a(x-h)^2 + k), you can complete the square.

Front

What is the vertex of a parabola represented by a quadratic equation?

Back

The vertex of a parabola represented by the quadratic equation y = ax^2 + bx + c is the point (h, k), where h = -\frac{b}{2a} and k is the value of the function at h.

Front

What is the axis of symmetry in a quadratic function?

Back

The axis of symmetry of a quadratic function is the vertical line x = -\frac{b}{2a}, which divides the parabola into two mirror-image halves.

Front

How do you solve a quadratic equation using the quadratic formula?

Back

To solve a quadratic equation using the quadratic formula, identify the coefficients a, b, and c, substitute them into the formula x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, and simplify to find the values of x.

Front

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

Back

The roots of a quadratic equation ax^2 + 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) = 0, where r1 and r2 are the roots.

Front

What does it mean for a quadratic equation to have one real solution?

Back

A quadratic equation has one real solution when the discriminant is zero (D = 0), indicating that the parabola touches the x-axis at exactly one point (the vertex).