MTH 161 Ch2 Quadratics and Factoring (2.1)

Three forms of a quadratic?

Standard: ax^2 + bx + c. Vertex: a(x-h)^2 + k. Factored: a(x-r)(x-s).

What does vertex form tell you?

The vertex is (h, k) and the axis of symmetry is x = h. It also shows the transformations from x^2.

What does the c in standard form tell you?

c is the y-intercept, the point (0, c).

How do you find the vertex from standard form?

x = -b / (2a), then plug that x back into the function to get y. The axis of symmetry is the same line x = -b / (2a).

What does the sign of a do?

Positive a opens up (vertex is a minimum). Negative a opens down (vertex is a maximum). Larger |a| means a narrower parabola.

Vertex, axis and intercepts of f(x) = (x-3)^2 - 4?

Vertex (3, -4), axis x = 3, x-intercepts x = 1 and x = 5, y-intercept (0, 5).

How do you find the x-intercepts of 2x^2 + 20x + 42?

Pull out the GCF first: 2(x^2 + 10x + 21) = 2(x+3)(x+7). Zeros at x = -3 and x = -7.

Vertex of 2x^2 + 20x + 42, and vertex form?

x = -20/4 = -5, y = -8, so vertex (-5, -8). Vertex form: 2(x+5)^2 - 8.

Domain and range of f(x) = -x^2 - 2x + 3?

Domain (-infinity, infinity). It opens down with vertex (-1, 4), so range (-infinity, 4].

What is the quadratic formula?

x = [-b plus or minus sqrt(b^2 - 4ac)] / (2a).

What is the discriminant?

b^2 - 4ac. Positive: two real solutions. Zero: one real solution. Negative: no real solutions.

What is the first step of any factoring problem?

Pull out the GCF. If x is in every term, factor it out too.

How do you factor ax^2 + bx + c when a is not 1?

Multiply a times c, find two numbers that multiply to that and add to b, split the middle term, then factor by grouping.

Difference of two squares?

a^2 - b^2 = (a+b)(a-b). Check for it as a final step, for example x^4 - 5x^2 + 4 = (x+2)(x-2)(x+1)(x-1).

Sum and difference of cubes?

a^3 + b^3 = (a+b)(a^2 - ab + b^2). a^3 - b^3 = (a-b)(a^2 + ab + b^2). Signs: sum is plus, minus, plus. Difference is minus, plus, plus.

How do you factor by grouping with four terms?

Group the first two and last two terms, pull the GCF from each group, then factor out the common binomial. If you factor out a negative GCF, adjust signs so the binomials match.