In rectangle RSTU, diagonals RT and SU intersect at O. If RT = 9x + 6 and SO = 10.5x - 9, what is the length of US?
In order to solve this equation, you need to know about the properties of rectangles. First of all, both of the diagonals of a rectangle are congruent. Also, the diagonals bisect each other, meaning that one diagonal cuts the other diagonal in half, and vice verse. RT is a diagonal, and SO is half of a diagonal. Therefore: RT = 2 (RT) Plus in the expressions given and solve for x: 9x + 6 = 2 (10.5x - 9) 9x + 6 = 21x - 18 -12x = -24 x = 2 The problem is asking for the length of US, in other words, the length of a diagonal. Because RT is also a diagonal, we can plug in x = 2 for the expression for RT, and that will be congruent to the length of US: SO = RT SO = 9x + 6 SO = 9(2) + 6 SO = 18 + 6 SO = 24 The length of US is 24.
Find the minimum and maximum points of the following function y = 3x^4 - 8x^3 - 144x^2
To calculate minimum and maximum points, we need to find where the first derivative is equal to zero, because that is where the slope of the tangent line will be a horizontal line, therefore form either a peak (maximum) or a valley (minimum). f(x) = 3x^4 - 8x^3 - 144x^2 f'(x) = 12x^3 - 24x^2 - 288x Using the power rule, we found the derivative of the given function. Now, we need to factor the derivative, and then set the equation equal to zero and solve for x. f'(x) = (12x) (x^2 - 2x - 24) f'(x) = (12x) (x - 6) (x + 4) 0 = (12x) (x - 6) (x + 4) 12x = 0 x - 6 = 0 x + 4 = 0 x = 0 x = 6 x = -4 Those 3 x-values are what are called critical values, meaning the x-values at which the minimums and maximums exists. To find their y-values, plug them into the ORIGINAL FUNCTION: f(-4) = -1024 f(0) = 0 f(6) = -3024 From that, we can determine that our critical points are (-4, 1024), (0,0), (6, -3024) Now to determine if a point is a minimum or a maximum, plug them into the second derivative. Don't worry about the number, just look for if the number is positive or negative: f'(x) = 12x^3 - 24x^2 - 288x f''(x) = 36x^2 - 48x - 288 f''(-4) = + f''(0) = - f''(6) = + A positive second derivative means concave up, forming a minimum, because the slopes will gradually become more and more positive. A negative second derivative means concave down, forming a maximum, because the slopes will gradually become more and more negative. Therefore, there is a minimum at (-4, 1024), a maximum at (0, 0), and a minimum at (6, -3024)
Factor: 4x^4 - 20x^3 - 14x^2 + 70x
The first thing you should look for in any factoring problem is a GCF (Greatest Common Factor). In other words, what combination of coefficients and variables can you divide into EVERY SINGLE term? In this problem, we can divide 2x: 2x (2x^3 - 10x^2 - 7x + 35) Next, check to see if the polynomial inside of the parenthesis can be factored. Let's try factoring by grouping: 2x [ (2x^3 - 10x^2) + (- 7x + 35) ] Now, factor what you can out of each individual set of parenthesis. KEEP IN MIND, the goal is to have one of the factors in the first term be identical to one of the factors in the second term: 2x [ (2x^2)(x - 5) + (-7)(x - 5) ] ***Notice how for the second term we factored out a (-7)? That's because if we would have factored out a (7), the second part would be (-x + 5). We want them to match, so I factored out a (-7) to make the second part (x - 5)*** The next step in factoring by grouping is to take the first part of each term and combine them to create their own term: 2x [ (2x^2 - 7)(x - 5) ] The answer is: 2x (2x^2 - 7) (x - 5)