I'm doing some word problems on finding the areas of rectangles, its just like all complicated. I don't understand how to do them. For example:
A square field had 3 m added to its length and 2m added to its width. The area is 90 m2(squared). Find the length of a side of the original field.
So first you get (x+3)(x+2)=90, right?
then you get x2+5x+6=90
then x2+5x=84
How do you go on? How do you figure out x? I don't get it! I have other problems like those, could you tell me how to figure out a problem with x2 and something x = a number in general? Please help!
Take your step:
x^2 + 5x + 6 = 90
Now:
x^2 + 5x - 84 = 0
Factor by inspection or use the quadratic formula to solve for x.
A hint is what two factors of 84 differ by 5?
To solve the equation x^2 + 5x = 84, you need to rearrange it into the form ax^2 + bx + c = 0. In this case, you have x^2 + 5x - 84 = 0.
To solve quadratic equations like this one, you can use various methods, such as factoring, completing the square, or using the quadratic formula. In this case, factoring is the most appropriate and efficient method.
To factor the quadratic equation, you need to find two numbers that multiply to give you -84 (the product of the coefficient of x^2 and the constant term) and add up to give you 5 (the coefficient of x). In this case, those numbers are 12 and -7.
So, you rewrite the equation as (x + 12)(x - 7) = 0.
Now, to find the value of x, you set each factor equal to zero and solve for x:
x + 12 = 0 or x - 7 = 0
Solving the first equation, x + 12 = 0, gives you x = -12.
Solving the second equation, x - 7 = 0, gives you x = 7.
Therefore, there are two possible values of x: -12 and 7.
However, in this context of finding the length of a side of the original field, a negative value does not make sense since length cannot be negative. Hence, the length of the side of the original field is 7 meters.