Chapter 8: Rational Functions
... 8.5 Solve Rational Equations Rational Equation A rational equation is two equal rational expressions. Steps to Solve a Rational Equation 1. Clear all denominators (i.e. make all denominators 1) in the equation by multiplying both sides of the equation by the LCD of all rational expressions. 2. Solve ...
... 8.5 Solve Rational Equations Rational Equation A rational equation is two equal rational expressions. Steps to Solve a Rational Equation 1. Clear all denominators (i.e. make all denominators 1) in the equation by multiplying both sides of the equation by the LCD of all rational expressions. 2. Solve ...
Solving Multi-Step Equations
... Step 2: Use the Distributive Property to remove parenthesis if you can’t simplify the problem within them. Example: 4 (25) = 4 (20 + 5) = 4 • 20 + 4 • 5 = 80 + 20 = 100 Step 3: Combine like terms on each side. Examples: 4x + 5x = 9x 8a – 5a = 3a Step 4: “Undo” (since you’re working backwards) additi ...
... Step 2: Use the Distributive Property to remove parenthesis if you can’t simplify the problem within them. Example: 4 (25) = 4 (20 + 5) = 4 • 20 + 4 • 5 = 80 + 20 = 100 Step 3: Combine like terms on each side. Examples: 4x + 5x = 9x 8a – 5a = 3a Step 4: “Undo” (since you’re working backwards) additi ...
Area of a Triangle
... polygon. Area is 2-dimensional like a carpet or an area rug. A triangle is a three-sided polygon. We will look at several types of triangles in this lesson. To find the area of a triangle, multiply the base by the height, and then divide by 2. The division by 2 comes from the fact that a parallelogr ...
... polygon. Area is 2-dimensional like a carpet or an area rug. A triangle is a three-sided polygon. We will look at several types of triangles in this lesson. To find the area of a triangle, multiply the base by the height, and then divide by 2. The division by 2 comes from the fact that a parallelogr ...
Test also includes review problems from earlier sections so study
... Prime (Not Factorable) - Sum of Squares [NOTE: If you got a different answer, carefully multiply it back together to see that it is not equal to the original problem.] ...
... Prime (Not Factorable) - Sum of Squares [NOTE: If you got a different answer, carefully multiply it back together to see that it is not equal to the original problem.] ...
Multistep Equations
... • Use the distributive property to eliminate parentheses • Ex) 5(x – 2) + x = 2(x + 3) simplifies to 5x – 10 + x = 2x + 6 + 4 ...
... • Use the distributive property to eliminate parentheses • Ex) 5(x – 2) + x = 2(x + 3) simplifies to 5x – 10 + x = 2x + 6 + 4 ...
Prealgebra Curriculum Outline 2011
... Understand place value as it relates to powers of 10 Multiply and dividing by powers of 10 Convert between standard form and scientific notation Simplify numerical expressions with exponents using the order of operations. Understand the difference between prime and composite numbers Determine the pr ...
... Understand place value as it relates to powers of 10 Multiply and dividing by powers of 10 Convert between standard form and scientific notation Simplify numerical expressions with exponents using the order of operations. Understand the difference between prime and composite numbers Determine the pr ...
The Diophantine equation x4 ± y4 = iz2 in Gaussian
... ax4 +by 4 = cz 2 in Gaussian integers with only trivial solutions were studied. In [2] a different proof than Hilbert’s, using descent, that x4 + y 4 = z 4 has only trivial solutions in Gaussian integers. In this short note, we will examine the Diophantine equation x4 ± y 4 = iz 2 in Gaussian intege ...
... ax4 +by 4 = cz 2 in Gaussian integers with only trivial solutions were studied. In [2] a different proof than Hilbert’s, using descent, that x4 + y 4 = z 4 has only trivial solutions in Gaussian integers. In this short note, we will examine the Diophantine equation x4 ± y 4 = iz 2 in Gaussian intege ...
Solution to Practice Questions
... Hint : Proceed as in the proof of Theorem 23.5.1, but consider m = 4p1 p2 · · · pn − 1. Sol. Suppose for a contradiction that there are only finitely many prime numbers which are congruent to 3 modulo 4, say p1 , p2 , . . . , pn . Consider the number m = 4p1 p2 · · · pn − 1. Then m is congruent to 3 ...
... Hint : Proceed as in the proof of Theorem 23.5.1, but consider m = 4p1 p2 · · · pn − 1. Sol. Suppose for a contradiction that there are only finitely many prime numbers which are congruent to 3 modulo 4, say p1 , p2 , . . . , pn . Consider the number m = 4p1 p2 · · · pn − 1. Then m is congruent to 3 ...
Notes on Solving Quadratic Equations by Factoring
... (Set factors equal to 0) (Solve each factor for x) Continued ...
... (Set factors equal to 0) (Solve each factor for x) Continued ...
Solutions
... of variables u = x + y and v = y and find an equation relating u and v. Then mimick how we found all Pythagorean triples.] Proof. Via the change of variables we get u2 + 2v 2 = x2 + 2xy + 3y 2 = 2 which has (0, 1) as a solution. If (u, v) 6= (0, 1) is another solution let t be the x-coordinate of th ...
... of variables u = x + y and v = y and find an equation relating u and v. Then mimick how we found all Pythagorean triples.] Proof. Via the change of variables we get u2 + 2v 2 = x2 + 2xy + 3y 2 = 2 which has (0, 1) as a solution. If (u, v) 6= (0, 1) is another solution let t be the x-coordinate of th ...
Sec 2.1 - studylib.net
... Throughout chapter one, we solved several types of equations including linear equations, quadratic equations, rational equations, etc. Each of these equations had something in common. They were all examples of equations in one variable. In this chapter, we will study equations involving two variable ...
... Throughout chapter one, we solved several types of equations including linear equations, quadratic equations, rational equations, etc. Each of these equations had something in common. They were all examples of equations in one variable. In this chapter, we will study equations involving two variable ...
Pythagorean triple
A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are coprime. A right triangle whose sides form a Pythagorean triple is called a Pythagorean triangle.The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula a2 + b2 = c2; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. For instance, the triangle with sides a = b = 1 and c = √2 is right, but (1, 1, √2) is not a Pythagorean triple because √2 is not an integer. Moreover, 1 and √2 do not have an integer common multiple because √2 is irrational.