Simplifying Square Root Radicals
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The presentation contains several examples showing how to simplify square root radicals using perfect square factors of the radicand.
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Simplifying Radicals
This document discusses simplifying radicals. It provides examples of perfect square factors and leaving expressions in radical form. It also covers combining like radicals, multiplying radicals by multiplying coefficients and radicands, and dividing radicals by dividing coefficients and radicands when possible and rationalizing the denominator to remove radicals. The document uses examples such as simplifying √48, √80, √50, and √125 to demonstrate these concepts.
Roots and radical expressions
This document discusses operations involving radicals such as square roots. Some key points covered include:
1) Simplifying radicals by pulling out perfect squares and simplifying inside and outside the radical.
2) Multiplying radicals by multiplying only the numbers outside the radicals while keeping numbers inside the same.
3) Rationalizing denominators by multiplying the denominator and numerator by the radical in the denominator to remove radicals from the denominator.
4) Adding and subtracting radicals by combining like terms where only the numbers outside the radicals are added/subtracted and numbers inside are kept the same.
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This document discusses operations involving radicals such as square roots. Some key points covered include:
1) Simplifying radicals by pulling out perfect squares and simplifying inside and outside the radical.
2) Multiplying radicals by multiplying only the numbers outside the radicals while keeping numbers inside the same.
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Okay, let's think through this with the new information:
* The equation modeling the height is: h = -16t^2 + vt + c
* The initial height (c) is still 2 feet
* The initial velocity (v) is now 20 feet/second
* The target height (h) is still 20 feet
So the equation is:
20 = -16t^2 + 20t + 2
0 = -16t^2 + 20t + 18 (subtract 20 from both sides)
Evaluating the discriminant:
(20)^2 - 4(-16)(-18) = 400 - 288 = 112
Since the discriminant is positive
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Okay, let's think through this with the new information:
* The equation modeling the height is: h = -16t^2 + vt + c
* The initial height (c) is still 2 feet
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* The target height (h) is still 20 feet
So the equation is:
20 = -16t^2 + 20t + 2
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1) A single point of intersection, meaning one solution.
2) Parallel lines, meaning no solution.
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The type of solution can be determined without graphing by writing the equations in slope-intercept form and comparing slopes and y-intercepts. Different slopes means one solution, same slope but different intercepts means no solution, and same slope and intercept means infinite solutions.
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This document provides steps for solving radical equations:
1) Isolate the radical on one side of the equation by performing inverse operations
2) Raise both sides of the equation to a power equal to the index of the radical to eliminate the radical
3) Solve the remaining polynomial equation
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In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalized version of the familiar arithmetic technique called long division. It can be done easily by hand, because it separates an otherwise complex division problem into smaller ones.
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https://en.wikipedia.org/wiki/Polynomial_long_division
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This document summarizes key concepts about quadratic equations, including:
- Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula.
- Completing the square involves manipulating the equation into a perfect square trinomial form.
- The quadratic formula provides the solutions to any quadratic equation in standard form.
- Cubic equations that are the sum or difference of cubes can be factored and solved.
- Literal quadratic equations can be solved for a specified variable using techniques like the square root property or quadratic formula.
- The discriminant determines whether the solutions to a quadratic equation are rational, irrational, or complex numbers.
1.4 Quadratic Equations
This document discusses methods for solving quadratic and cubic equations. It begins by introducing quadratic equations in standard form and methods for solving them, including factoring, completing the square, and using the quadratic formula. It then discusses properties related to the square root and applies them to solving quadratic equations. The document concludes by introducing cubic equations that are the sum or difference of cubes, and provides an example of solving one using factoring.
Quinary base five
The document summarizes conversions between decimal and quinary (base 5) number systems. It provides examples of adding, subtracting, multiplying, and dividing numbers in quinary notation. For each operation, it shows the step-by-step work and explains how to "carry" or borrow in base 5. Common addition, subtraction, multiplication, and division facts are also presented in quinary as examples.
Adding & subtracting rational expressions
This document provides an overview of adding and subtracting rational expressions. It begins with instructions on finding the least common denominator (LCD) and provides examples of adding fractions with unlike denominators. It then demonstrates subtracting fractions with unlike denominators by first making the denominators the same. Several examples of adding and subtracting rational expressions are worked through step-by-step. Special cases involving factoring denominators are also discussed.
Radicals are radical
This document provides instructions for working with radicals and irrational square roots, including:
1) How to simplify irrational square roots by looking for perfect squares within numbers.
2) How to add radicals by expressing them with the same radicand and using the distributive property.
3) How to subtract radicals by expressing them with the same radicand and using the distributive property.
The document uses examples such as √50 = 5√2 and 4√7 + 5√7 = 9√7 to demonstrate techniques for simplifying radicals.
Q1-W1-Factoring Polynomials.pptx
This document provides instruction on factoring polynomials. It covers several factoring methods including common monomial factoring, difference of squares, sum and difference of cubes, perfect square trinomials, and general trinomials. Examples are provided for each method. The objectives are to determine appropriate factoring methods, factor polynomials completely using various techniques, and solve problems involving polynomial factors.
Surds.pptx
1) A surd is a number whose square root is not a whole number. Common surds include √2, √3, √5.
2) Surds can be simplified by breaking numbers into factors where one is a perfect square.
3) Surds can be added, subtracted, multiplied, or divided following specific rules such as having the same basic form or multiplying by conjugates.
Quadratic Equation
This document introduces methods for solving quadratic equations beyond factoring, including the square root property, completing the square, and the quadratic formula. It discusses how to determine the number and type of solutions based on the discriminant. The key steps are presented for solving quadratics, graphing quadratic functions as parabolas, and finding the domain and range. Piecewise-defined quadratic functions are also explained.
Adding and subtracting radicals ppt white board practice
This document contains a series of practice problems for adding and subtracting radical expressions. It provides the problems, step-by-step workings, and answers. There are 18 questions in total covering topics like combining like terms under a single radical, simplifying radicals, and identifying errors in working problems. The document emphasizes studying for an upcoming quiz on radicals.
Orderofoperations 201217015300
The document discusses using the order of operations to evaluate expressions. It provides the mnemonic "Please Excuse My Dear Aunt Sally" (PEMDAS) to remember the order: Parentheses, Exponents, Multiply/Divide (left to right), Add/Subtract (left to right). Several examples are worked through step-by-step to demonstrate evaluating expressions according to this order. The document also covers evaluating expressions when variables are substituted with values.
Solving quadratic equation using completing the square
1. The document provides steps for solving quadratic equations using completing the square. It involves isolating terms with x, finding the square of half the coefficient of x, completing the square of the left side of the equation, applying the square root property, and solving for the roots.
2. Three examples are shown applying these steps: finding the solution set of x^2 - 18x = -17 to be x = 17 and x = 1; solving x^2 + 10x - 11 = 0 to get x = 1 and x = -11; and solving x^2 = -24 + 10x to get x = 6 and x = 4.
Solving Quadratic-Equation.pptx
The document provides information about an online math class, including:
- A prayer asking God for guidance and wisdom as students wait to be taught.
- Reminders for online class such as turning on cameras, being on time, muting/unmuting audio.
- The weekly task of answering pretest questions from the module.
- An overview of the math topic for the first week of quarter 1 - quadratic equations. It discusses methods for solving quadratic equations such as extracting square roots, factoring, completing the square, and using the quadratic formula.
Question and Solutions Exponential.pdf
Unlock a deep understanding of mathematics with our Module and Summary! Clear definitions, comprehensive discussions, relevant example problems, and step-by-step solutions will guide you through mathematical concepts effortlessly. Learn with a systematic approach and discover the magic in every step of your learning journey. Mathematics doesn't have to be complicated—let's make it simple and enjoyable!
1.3 solving equations t
The document provides examples and explanations for solving different types of equations, including:
1) Polynomial equations through factoring or the quadratic formula.
2) Rational equations by clearing denominators.
3) Radical equations by squaring both sides to remove radicals.
4) Absolute value equations by recognizing that |x-c|=r implies x=c±r.
The document also discusses solving power equations, finding zeros and domains of functions, and using properties of absolute values.
5 chap
1) The document provides examples and formulas for calculating square roots and cube roots. It gives the definitions of square roots and cube roots and shows how to evaluate them using prime factorizations or long division methods.
2) Several word problems are worked out step-by-step to demonstrate how to find the square root or cube root of various numbers.
3) Formulas are provided for operations involving square roots, such as multiplying or dividing them.
9.5 Nonlinear Systems of Equations
This document discusses solving nonlinear systems of equations in three ways: elimination, substitution, and graphing. It provides examples of solving systems by each method and discusses key steps like eliminating variables, substituting values, and finding the intersection point(s) of graphs. Absolute value equations are also addressed. Students are expected to be able to solve nonlinear systems by any method, as questions may require non-graphical solutions for irrational answers. A classwork and quiz assignment is provided to practice these skills.
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