Unit 4-2 Assessment Review: Solving Quadratics A.7B, A.8A, A.10E

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 standard form of a quadratic equation is $$y = ax^2 + bx + c$$ .

The discriminant, given by $$D = b^2 - 4ac$$ , determines the nature of the roots: if D > 0, there are two distinct real roots; if D = 0, there is one real root; if D < 0, there are no real roots.

The vertex can be found using the formula $$x = -\frac{b}{2a}$$ . The y-coordinate can be found by substituting this x-value back into the equation.

The axis of symmetry is a vertical line that divides the parabola into two mirror images, given by the equation $$x = -\frac{b}{2a}$$ .

The maximum or minimum value occurs at the vertex. If a > 0, the parabola opens upwards and has a minimum; if a < 0, it opens downwards and has a maximum.

To solve by factoring, express the quadratic in the form $$(px + q)(rx + s) = 0$$ , then set each factor to zero and solve for x.

The quadratic formula is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ , used to find the roots of a quadratic equation.

Completing the square involves rewriting a quadratic in the form $$y = a(x - h)^2 + k$$ , where (h, k) is the vertex of the parabola.

The maximum height can be found by determining the vertex of the quadratic equation, which gives the time at which the maximum height occurs.