Textbook Question
Solve each equation using the quadratic formula. See Examples 5 and 6. 1/2x^2 + 1/4x - 3 = 0
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1. Equations & Inequalities
The Square Root Property
2:28 minutesProblem 73
Textbook Question
Solve each equation for the specified variable. (Assume no denominators are 0.) See Example 8. F = kMv^2/r , for v
Verified step by step guidance1
Identify the equation given: \( F = \frac{kMv^2}{r} \).
To solve for \( v \), first isolate the term containing \( v^2 \) by multiplying both sides by \( r \): \( Fr = kMv^2 \).
Next, divide both sides by \( kM \) to further isolate \( v^2 \): \( \frac{Fr}{kM} = v^2 \).
To solve for \( v \), take the square root of both sides: \( v = \sqrt{\frac{Fr}{kM}} \).
Remember that taking the square root introduces a \( \pm \) sign, so \( v = \pm \sqrt{\frac{Fr}{kM}} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Algebraic Manipulation
Algebraic manipulation involves rearranging equations to isolate a specific variable. This process includes operations such as addition, subtraction, multiplication, and division, applied to both sides of the equation to maintain equality. Understanding how to manipulate equations is crucial for solving for a variable, as it allows one to express the variable in terms of others.
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Variables and Constants
In algebra, variables represent unknown values and can change, while constants are fixed values. In the equation F = kMv^2/r, F, k, M, and r are constants, while v is the variable we need to solve for. Recognizing the roles of variables and constants helps in understanding how to isolate the variable of interest effectively.
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Quadratic Equations
The equation F = kMv^2/r can be rearranged into a quadratic form when solving for v, specifically v^2 = (Fr)/(kM). Quadratic equations are polynomial equations of the form ax^2 + bx + c = 0, and they can have zero, one, or two solutions. Familiarity with the properties of quadratic equations, including factoring and the quadratic formula, is essential for finding the solutions for v.
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