Question

Quadratics

Answer 1

A quadratic expression or equation is a polynomial of degree 2 (highest power of variable is 2).

Standard form: ax^2+bx+c where a, b and c are constants.

If ax^2+bx+c=0, solution can be found using formula: x=(-b±√(b^2-4ac))/2a which can be derived by completing the square:

x^2+bx/a+c/a=0; x^2+bx/a=-c/a; x^2+bx/a+b^2/4a^2=b^2/4a^2-c/a; (x+b/2a)^2=b^2/4a^2-c/a.

Square root each side: x+b/2a=±√(b^2/4a^2-c/a)=±√(b^2-4ac)/2a.

x=(-b±√(b^2-4ac))/2a.

Graph of a quadratic is a parabola (y=ax^2+bx+c) which resembles a U shape with broadening arms when a is positive. It can also be an upside down U (a is negative). The vertex is the lowest or highest point on the curve. Parabola can also be lying on its side: x=ay^2+by+c where y is the vertical axis and x the horizontal axis.

Answer 2

A quadratic equation follows the form of ax^2 + bx + c =0, and a quadratic function is a polynomial that follows that form. If a quadratic function equals 0, then it is a quadratic equation.

There are two ways solve quadratic equations. one is by factoring (which doesn't work for all problems) and the other is by using the quadratic formula (which sometimes takes longer than factoring).

For example, x^2 + 3x + 2 = 0 can be solved by:

a) Factoring

x^2 + 3x + 2 = (x+1)(x+2)

since x^2 + 3x + 2 = 0, then (x+1)(x+2) = 0

Then you separate the two parts in parentheses and solve for x.
x + 1 = 0; x = -1
x + 2 = 0; x = -2

If you can see that an equation can be factored easily, then this is the faster way to do it.

b) Quadratic Formula

The quadratic formula is (-b plus or minus the square root of (b^2 - 4ac))/2a

It looks kind of messy here; if you google quadratic formula it will pop up with some images.

So then you take the corresponding numbers from your equation and plug them into the formula.

a = 1 since x^2 has the coefficient of 1, b = 3 since x has coefficient of 3, and 2 since that is the number by itself. Plug these in.

( -3 plus or minus the square root of (3^2 - 4(1)(2)))/2(1)
(-3 plus or minus the square root of (9-8))/2
(-3 plus or minus 1)/2

Then you do both (-3 + 1)/2 and (-3 - 1)/2, getting -1 and -2.

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