Question

Write a quadratic equation in factored form to model the given problem. Be sure to write the entire equation.

The height of a triangle is 4 more than the base, and the area of the triangle is 6 square units. Find the length of the base. Let x = the length of the base.

Answer 1

i can walk you through this one.

Answer 2

what is the equation for the area of a triangle?

Answer 3

well thats my problem, i'm not sure how to write an equation for a triangle... i know x = 2 though

Answer 4

How do you know that x = 2?

Answer 5

Do you know the area of a rectangle?

Answer 6

because the base is x, the height is x+4 , and the area is 6, so by the basexheight/2 = area

Answer 7

2 * 6 \2 =6

Answer 8

Yes, the area of a triangle is base x height /2. That's what I was asking for that you said you didn't know

Answer 9

So that's the first step. We'll say that your base is b, and your height is h. Therefore the area of your triangle = b*h/2

Answer 10

But they tell you that the height is 4 more than the base. You write that in math as: h= 4+b

Answer 11

They also tell us that the area which we will call A is 6. So now let's put that all together

Answer 12

A=b*h/2 (area of a triangle)
A=6
Therefore b*h/2 = 6
But h=4+b therefore let's substitute for h in the equation above
b*(4+b)/2 = 6
If we expand it, we get:
( b*4 + b*b)/2 =6
To remove 2 from the denominator, we multiply both sides by 2
2*( b*4 + b*b)/2 =6*2
The 2s on the left side cancel out
( b*4 + b*b) =12
4b + b^2=12
Subtract 12 from both sides
4b + b^2 -12 = 0
Rearrange
b^2 + 4b -12 = 0
That's your quadratic equation

Answer 13

But they ask for you to write it in factored form, so you have to factor b^2 + 4b -12 = 0
You do that by finding the two values that will add together to give you +4b and multiply together to give you -12b^2.
those two values are +6b and -2b
[Check: +6b + (-2b) = +4b, and +6b*-2b = -12b^2 ]

So we factor
b^2 + 4b -12 = 0
b^2 -2b + 6b -12 = 0
b(b-2) + 6(b-2) = 0
(b-2)(b+6) = 0
that means (b-2) = 0 or (b+6) = 0
If b-2 = 0, therefore b = 2
if b+6 = 0, therefore b = -6
You get 2 answers, but you can't really have a "negative" length, so we know that b =2 is the right answer.

that's how you get the 2.

Answer 14

Oh by the way, the question asks you to says that the base = x (not b), so everywhere you see b, please replace with x
lol