Questions and Answers
What is the factor of $9y^2 + 36y + 35$?
The equation $6/x + 14 = 40$ represents a linear function.
False
What is the solution to the equation $3(2y+3) = 3-2(y-3)$?
0
What is the expression for two less than three times the sum of a number and four?
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What are the two numbers if one is 3 more than twice the other and their sum is 27?
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What is the result of multiplying $(9√7 - 4√6)(√7 + 3√6)$?
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What is the result when typing $(9-8i) - (-4 + 6i)$ in the form $a + bi$?
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What is the result of writing $√-1 + 7√-9$ in $a + bi$ form?
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What is the equation representing that the product of a number and 5 is 4 less than the quotient of that number and 2?
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What are the dimensions of a rectangle with a perimeter of 108m, where the length is 6 meters longer than twice the width?
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What is the product of the complex number $5 + 3i$ and its conjugate?
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What are the solutions to the equation $5x^2 + 18x - 8 = 0$ using the quadratic formula?
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What is the discriminant of the equation $2x^2 - 16x + 32 = 0$ and how many solutions does it have?
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What is the positive number when the sum of its square and the square of 2 more than the number equals 130?
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Study Notes
Algebra Concepts and Operations
- Factoring: The expression 9y² + 36y + 35 factors to (3y + 7)(3y + 5).
- Linear vs Non-linear: The equation 6/x + 14 = 40 is classified as non-linear. An equation like y = 2x - 3 is linear, while y = x² + x + 4 is non-linear, forming a parabola due to the exponent on x being greater than one.
Solving Equations
- To solve the equation 3(2y + 3) = 3 - 2(y - 3), the solution is y = 0.
- In expressions like 3 less than three times the sum of a number and four, it translates to the algebraic expression 3(x + 4) - 2.
- For the problem stating that one number is 3 more than twice another, if their sum is 27, the two numbers are determined to be 8 and 19.
Complex Numbers
- To multiply (9√7 - 4√6)(√7 + 3√6), the result is 23√42 - 9.
- Simplifying the expression (9 - 8i) - (-4 + 6i) results in the complex number 13 - 14i.
- The expression √-1 + 7√-9 translates to 7√2i in a + bi form.
Equations and Geometrical Concepts
- The relationship stating the product of a number and 5 being equal to 4 less than the quotient of that number divided by 2 yields the equation 5x = x/2 - 4.
- For a rectangle with a perimeter of 108m, where the length is 6 meters longer than twice the width, the dimensions result in width of 16m and length of 38m.
Complex Conjugates and Quadratic Formula
- The complex conjugate of 5 + 3i is 5 - 3i, and their product equals 34.
- Using the quadratic formula on the equation 5x² + 18x - 8 = 0 yields solutions of -1 - i√39 / 10 and -1 + i√39 / 10.
- For the quadratic equation 2x² - 16x + 32 = 0, the discriminant is 0, indicating there is one real solution.
Additional Calculations
- The solution to the sum of the square of a positive number and the square of two more than that number equating to 130 is the number 7.
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Test your knowledge with these practice problems in college algebra. This quiz covers factoring quadratic expressions and distinguishing between linear and non-linear equations. Use flashcards to strengthen your understanding of algebraic concepts.
