6.01 Solving Quadratics Review

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

The discriminant, given by $$D = b^2 - 4ac$$ , indicates the nature of the roots of a quadratic equation: if D > 0, there are two distinct real roots; if D = 0, there is one real root; if D < 0, there are two complex roots.

To factor a quadratic equation of the form $$ax^2 + bx + c$$ , find two numbers that multiply to $$ac$$ and add to $$b$$ , then express the equation as $$a(x - p)(x - q)$$ where p and q are the roots.

Completing the square is a method used to solve quadratic equations by rewriting the equation in the form $$(x - p)^2 = q$$ , making it easier to solve for x.

The vertex form of a quadratic equation is $$y = a(x - h)^2 + k$$ , where (h, k) is the vertex of the parabola.

The axis of symmetry of a quadratic function $$y = ax^2 + bx + c$$ is the vertical line given by $$x = -\frac{b}{2a}$$ .

A quadratic can be factored if its discriminant is a perfect square, indicating that the roots are rational numbers.

The standard form of a quadratic equation is $$ax^2 + bx + c = 0$$ , where a, b, and c are constants.

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