Factoring to solve quadratic equations: Find the value of x as it relates to the triangle.

Area = 600 cm^2 There is a large triangle that is divided by a 90 degree angle in the middle. That 90 degree angle line is "x" and the complete bottom of the large triangle is "2x +2". What is the value of "x"?

I hope this is not too confusing. Please solve and explain what I need to do. Thanks.

That line with the "x" on it is the height of the triangle.

A = 1/2bh
600 = 1/2(2x + 2)(x)
600 = (x + 1)(x)

To find the value of "x" in this triangle problem, we will need to use the information given and apply the concept of factoring to solve the quadratic equation.

First, let's break down the problem:

Area of the triangle = 600 cm^2
The triangle is divided by a line with a 90-degree angle, which we will represent as "x."
The length of the complete bottom of the triangle is represented as "2x + 2."

To solve for "x," we need to set up an equation based on the area of the triangle.

The formula to calculate the area of a triangle is:

Area = (base * height) / 2

In this case, the base of the triangle is "2x + 2," and the height is "x."

Plugging in the given area, we have the equation:

600 = (2x + 2) * x / 2

To simplify the equation, we can multiply through by 2 to get rid of the fraction:

1200 = 2x^2 + 2x

Now, let's rearrange the equation to form a quadratic equation:

2x^2 + 2x - 1200 = 0

Since this equation cannot be factored easily, we can use the factoring by grouping method.

Step 1: Multiply the leading coefficient and the constant term: 2 * (-1200) = -2400.

Step 2: We need to find two factors of -2400 that add up to the coefficient of "x" (which is +2).

The two factors we can use are +50 and -48 because 50 - 48 = 2.

Step 3: Now we rewrite the middle term using these two factors:

2x^2 + 50x - 48x - 1200 = 0

Step 4: Group the terms:

(2x^2 + 50x) + (-48x - 1200) = 0

Step 5: Factor out the greatest common factor from each group:

2x(x + 25) - 48(x + 25) = 0

Step 6: Now, factor out the common binomial:

(2x - 48)(x + 25) = 0

Using the Zero Product Property, to find x, we set each factor equal to zero:

2x - 48 = 0 or x + 25 = 0

Solving these equations individually gives us two possible values for x:

2x - 48 = 0
2x = 48
x = 24

x + 25 = 0
x = -25

Therefore, the value of "x" is either 24 or -25.

However, in the context of this triangle problem, the length of a line cannot be negative, so the only valid value for "x" is 24 cm.

Hence, the value of "x" in this triangle is 24 cm.