Quadratic Formula and Discriminant Review

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 b^2 - 4ac, indicates the nature of the roots of the quadratic equation. If it is positive, there are two distinct real solutions; if it is zero, there is one real solution; and if it is negative, there are two complex 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 zero discriminant?

Back

A zero discriminant indicates that the quadratic equation has exactly one real solution, meaning the parabola touches the x-axis at one point.

Front

How can 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, calculate the discriminant, and then apply the formula x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

Front

What is the first step in using the quadratic formula to solve an equation?

Back

The first step is to rearrange the equation into the standard form ax^2 + bx + c = 0, if it is not already in that form.

Front

What does it mean if a quadratic equation has two real solutions?

Back

It means that the graph of the quadratic function intersects the x-axis at two distinct points.

Front

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

Back

The sum of the roots is given by -b/a and the product of the roots is given by c/a, according to Vieta's formulas.

Front

How do you determine if a quadratic equation has complex solutions?

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If the discriminant (b^2 - 4ac) is negative, the quadratic equation has complex solutions.

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 can be found at the point (h, k), where h = -\frac{b}{2a} and k = f(h).