Quadratic Solutions

A quadratic equation is a polynomial equation of degree 2, typically in the form $$ax^2 + bx + c = 0$$ , where a, b, and c are constants and a ≠ 0.

The zeros of a quadratic function are the values of x that make the function equal to zero. They can be found using factoring, completing the square, or the quadratic formula.

The quadratic formula is used to find the solutions (or zeros) of a quadratic equation and is given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ .

The number of solutions can be determined by calculating the discriminant, $$D = b^2 - 4ac$$ . If D > 0, there are 2 real solutions; if D = 0, there is 1 real solution; if D < 0, there are no real solutions.

If a quadratic equation has no real solutions, it means that the graph of the quadratic function does not intersect the x-axis, which occurs when the discriminant is negative.

The roots of a quadratic equation can be found by identifying the x-intercepts of its graph. These are the points where the graph crosses the x-axis.

The vertex of a quadratic function is the highest or lowest point on the graph, depending on the direction of the parabola. It can be found using the formula $$x = -\frac{b}{2a}$$ .

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves and can be found using the formula $$x = -\frac{b}{2a}$$ .

The leading coefficient (a) determines the direction of the parabola: if a > 0, the parabola opens upwards; if a < 0, it opens downwards.

To solve by factoring, rewrite the quadratic in the form $$(x - p)(x - q) = 0$$ , where p and q are the roots, and then set each factor equal to zero.