Discriminant & Quadratic Formula

Front

What is the discriminant in a quadratic equation?

Back

The discriminant is the part of the quadratic formula under the square root, given by the expression b² - 4ac, where a, b, and c are coefficients of the quadratic equation ax² + bx + c = 0.

Front

How do you determine the number of solutions based on the discriminant?

Back

If the discriminant is positive, there are two distinct real solutions. If it is zero, there is exactly one real solution (a repeated root). If it is negative, there are no real solutions.

Front

What is the quadratic formula?

Back

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

Front

What does it mean if the discriminant is negative?

Back

If the discriminant is negative, the quadratic equation has no real solutions, meaning the parabola does not intersect the x-axis.

Front

What does it mean if the discriminant is zero?

Back

If the discriminant is zero, the quadratic equation has exactly one real solution, which is a repeated root, meaning the parabola touches the x-axis at one point.

Front

What is the significance of the roots of a quadratic equation?

Back

The roots (or solutions) of a quadratic equation are the values of x where the equation equals zero, representing the x-intercepts of the corresponding parabola.

Front

How can you find the roots of a quadratic equation using factoring?

Back

To find the roots by factoring, rewrite the quadratic equation in the form (x - r1)(x - r2) = 0, where r1 and r2 are the roots, and solve for x.

Front

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

Back

For a quadratic equation ax² + bx + c = 0, the sum of the roots is -b/a and the product of the roots is c/a.

Front

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

Back

The vertex of a parabola given by the equation y = ax² + bx + c can be found at the point (h, k), where h = -b/(2a) and k is the value of the function at h.

Front

How do you graph a quadratic function?

Back

To graph a quadratic function, identify the vertex, axis of symmetry, and intercepts, then plot points and draw a smooth curve through them.