Question

Use a diagram to show that (3x+1)(x+2) is equivalent to 3x2+7x+2.

Answer 1

3x*x = 3x^2

3x*2 = 6x

x*1 = x

1*2 = 2

that's basically the FOIL method, multiply first terms, the x terms
then the outside terms, the the inside terms, then the constant (last) terms
F=first, O=outside, I = Inside, L = last

add them up to get 3x^2 +6x +x + 2 = 3x^2 +7x +2

(3x+1)(x+2) = 3x^2 +7x+2

draw a rectangle with one side as 3x+1, the other side as x+2. The area of the rectangle is their product

divide the rectangle into 4 smaller rectangles. with sides 3x by x, 3x by 2, x by 1 and 2 by 1
calculate the areas of each rectangle the sum them.

standard form is

y=ax^2 +bx + c or

f(x) = x^2 +7x +2 where a=1, b=7, c=2

another standard form is to put the quadratic equation in vertex form
by completing the square

f(x) or y = x^2 +7x + 49/4 = -2 + 49/4

y = (x+3/2)^2 = 41/4

where the vertex = (3/2,41/4)

standard equation in vertex form is

y= a(x-h)^2 + k where (h,k) = vertex = (3/2,41/4) with a=1

It's an upward opening parabola with y intercept = 41/4 = 10 1/4, axis of symmetry x=3/2