Algebra: Quadratic Equations & Parabolas

ax^2 + bx + c = 0 , with a \neq 0

x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}

\Delta = b^2 - 4ac , the part under the square root in the quadratic formula. Its sign tells you how many real roots there are.

The parabola crosses the x-axis twice.

The vertex touches the x-axis.

It has two complex conjugate roots, and the parabola never touches the x-axis.

y = a(x - h)^2 + k , with vertex (h, k)

x = -\dfrac{b}{2a}

a > 0 : opens up, vertex is a minimum. a < 0 : opens down, vertex is a maximum.

y = a(x - r_1)(x - r_2) , where r_1 and r_2 are the x-intercepts

Sum: r_1 + r_2 = -\dfrac{b}{a} Product: r_1 r_2 = \dfrac{c}{a}

(x - 2)(x - 3) = 0 , so x = 2 or x = 3

x = \pm 3

a^2 - b^2 = (a + b)(a - b)

a^2 + 2ab + b^2 = (a + b)^2 a^2 - 2ab + b^2 = (a - b)^2

\left(\dfrac{b}{2}\right)^2 , which gives \left(x + \dfrac{b}{2}\right)^2

x^2 + 6x + 9 = 4 \Rightarrow (x + 3)^2 = 4 \Rightarrow x + 3 = \pm 2 x = -1 or x = -5

25

(2x - 1)(x + 2) = 0 , so x = \tfrac{1}{2} or x = -2

x = -\dfrac{-4}{2} = 2 , y = 4 - 8 + 1 = -3 Vertex: (2, -3)

\Delta = 4 - 20 = -16 x = \dfrac{-2 \pm 4i}{2} = -1 \pm 2i

(0, c)

Vertex (1, 3) , opens up, minimum value 3, axis of symmetry x = 1 . It has no x-intercepts.

For y = ax^2 + bx + c they all refer to the x-values where y = 0 . Only real roots show up as x-intercepts on the graph.

Factoring: when integer factors are easy to spot Square root method: when there is no bx term Completing the square: when you need vertex form Quadratic formula: always works

t = -\dfrac{64}{2(-16)} = 2 seconds h(2) = -64 + 128 = 64 feet