Intermediate Algebra Summary - Part II
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1 Intermediate Algebra Summary - Part II This is an overview of the key ideas we have discussed during the middle part of this course. You may find this summary useful as a study aid, but remember that the only way to really master the skills and understand the ideas is to practice solving problems. The Vertex Formula Example: Find the coordinates of the vertex of the parabola y 2x 2 + 2x 3. The x-coordinate is given by the formula x v b 2a, so x v 2 2(2) 3. Plug this back into the equaion of the parabola to find the corresponding y-coordinate: y v 2( 3) 2 + 2( 3) Therefore the coordinates of the vertex are ( 3, 2). Example: The profit for a company that sells x Things is P(x) 0.05x x 00 dollars. Determine how many Things the company should sell to make a maximum priofit, and what that profit will be. The maximum corresponds to the vertex of the parabola we get when e graph the function. The x-coordinate of that vertex is x v b 2a 30 2( 0.05) 300. So the company should sell 300 Things to get the maximum profit. In that case, the profit will be p(30) 0.05(300) (300) dollars. Quadratic Equations From Word Problems Example: A company s profit is given by the formula P(x) 0.02x x 0, 000 dollars, where x is the number of units made and sold. Find the break-even values for the company. Round to the nearest integer. Example: The company will break even when the proift is zero dollars, so we need to solve the equation 0.02x x 0,
2 We will solve this using the quadratic formula, with a 0.02, b 50, c 0000: x b ± b 2 4ac 2a 50 ± (50) 2 4( 0.02)( 0, 000) 2( 0.02) 50 ± ± 700 So Hence x or x x 29.2 or x Rounding to the nearest integer gives x 29 or x 228. Example: A boat travels north at maximum speed for hour. Then it travels east at maximum speed for 3 hours. After that time, it is 200 miles from it s starting point. What is the maximum speed of the boat? Give an answer in miles per hour, rounded to one decimal place. We let x represent the speed of the boat in miles per hour. Then after one hour, it will have travelled x miles north. After three hours of heading east, it will have travelled 3x miles in that direction. This is illustrated in the following figure: 3x x 200 We xcan now use the Pythagorean Theorem to find an equation that will allow us to solve for the unknown x: x 2 + (3x) 2 (200) 2 x 2 + 9x x x x ± But since x must be positive in this context, we see that the solution is x So the boat travels at approximately 63.2 miles hour.
3 Reducing Rational Expressions to Lowest Terms Example: Reduce x 2 4 x 2 +3x+2 Example: Reduce x2 +x+30 x 2 7x x 2 + x + 30 x 2 7x x 2 4 (x + 2)(x 2) x 2 + 3x + 2 (x + 2)() x 2 2x2 0x x x2 0x x 2 25 Example: Reduce x2 +4x x 2 2x x2 2x 24 x+ x 2 + 4x x 2 2x x2 2x 24 Example: Write x 2 2 x 2 +2x+ (x + 5)(x + 6) 2x(x 5) x(x 7) (x + 5)(x 5) 2(x + 6) x 7 22 x 7 x(x + 4) x(x 2) (x + 4)(x 6) (x 2)(x 6) x 2 82 as a single fraction. x 2 2 x ()(x ) 2 ()(x + 2) (x + 2) ()(x )(x + 2) 2(x ) ()(x + 2)(x ) (x + 2) 2(x ) ()(x + 2)(x ) x + 2 2x + 2 ()(x + 2)(x ) 4 x ()(x + 2)(x )
4 Solving Rational Equations Example: Solve the equation x+ 2. We begin by multiply both sides by (x)() in order to clear all the fractions: [ (x)(x ) ] 2(x)() Distribute on the left side: Cancel where appropriate: (x)() x + (x)() 2x() () + x 2x() Now simplify and solve the resulting equation for x: 2 2x 2 + 2x 2x 2 2 x2 x ± 2 Variation Example: When x 2, we have y 8. When x 4, we have y 2. Classify the type of variation expressed by this data, and use that classification to predict the value of y when x 8. Note that a larger value of x corresponds to a smaller value of y, so we only need to check inverse and inverse-square variation. The table below indicates the values of the variation constant we get for each of these options. Recall that for inverse variation, k xy, and for inverse-square variation, k x 2 y. x y xy x 2 y These calculation indicate that the variation is inverse square, with constant of variation k 32. Therefore, the equation that relates x and y is Consequently, when x 8, we must have y 32 x 2. y 32 (8)
5 Root Functions Example: Simplify the expression Note that 90 (2)(3)(3)(5) and 000 (2)(2)(2)(5)(5)(5). Therefore (2)(3)(3)(5) + (2)(2)(2)(5)(5)(5) 3 (2)(5) + (2)(5) (2)(5) Example: What is the domain of the function f(x) 3 2x? The square-root function is only defined when the argument inside of it is nonnegative, so we need to have 3 2x 0 Hence 3 2x so 3 2 x. That is to say, the domain of the function f(x) is x 3 2.
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