Podcast
Questions and Answers
What is the first step in solving an equation by completing the square?
What is the first step in solving an equation by completing the square?
To isolate the quadratic term and the linear term on one side of the equation.
When factoring a polynomial fully, why is it essential to look for a greatest common factor (GCF)?
When factoring a polynomial fully, why is it essential to look for a greatest common factor (GCF)?
The GCF simplifies the polynomial and makes it easier to factor further.
What do you need to remember when factorizing an expression involving a difference of squares?
What do you need to remember when factorizing an expression involving a difference of squares?
The difference of squares can be factored into the product of two binomials: $a^2 - b^2 = (a - b)(a + b)$.
How can you determine if a quadratic expression can be factored into rational numbers?
How can you determine if a quadratic expression can be factored into rational numbers?
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What must you add to both sides of an equation to complete the square?
What must you add to both sides of an equation to complete the square?
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In the process of completing the square, why do you divide the linear term by 2 before squaring it?
In the process of completing the square, why do you divide the linear term by 2 before squaring it?
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What is the significance of a polynomial being expressed in fully factored form?
What is the significance of a polynomial being expressed in fully factored form?
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What does the vertex form of a quadratic function indicate about its graph?
What does the vertex form of a quadratic function indicate about its graph?
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Study Notes
Algebra Problems
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Simplify: 3m²n × 4mn³ / (6mn)² = 2mn²
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Simplify: √18 - √54 + √50 - √24 = √2 + √2 = 2√2
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Solve: 2y² + 7y - 15 = 0
- y = (-7 ± √(7² - 4 * 2 * -15)) / (2 * 2)
- y = (-7 ± √169) / 4
- y = (-7 ± 13) / 4
- Two solutions: y = 3/2 and y = -5
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Factorise: a² - 9b² + 4a - 12b = (a - 3b)(a + 3b) + 4(a - 3b) = (a - 3b)(a + 3b + 4)
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Simplify: 6p² + 3pq / 8pq + 4q² = (3p + q)(2) / (2q)(2p + q)
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Solve: 2⁵x × 8x³ = 16x³ + 3
- (Not an appropriate equation for an immediate solution process, needs further clarity or retyping)
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Rationalise: √5 - 1 = √5-1 / (√5 -1) *(√5 + 1) / (√5 + 1)
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Simplify: (x² - 16) / (x² - 2) = (x-4)(x+4)/(x-2)(x+1) (Not fully simplified )
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(This solution is incomplete, further information or clarifications is required)
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Show:√2 - 1 + 4 / √2 + 1 = √2+3/ √2 + 1 = √2 +3 - √2 +1
- (Not a valid solution ; this is not an algebraic expression it is a numerical problem)
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Find: 4 / (3 - √5) = a + b√5 -Rationalize the denominator : 4 / (3 - √5) x (3+ √5 / 3+ √5) = (12 + 4√5)/ (9-5) = (12+4√5 / 4)
= 3 + √5 Then a = 3 and b = 1 -
Simplify (Indices): a⁻¹ + b⁻¹ / a + b = 1/a + 1/b /(a+b)= (a+b)/ ab/ a+b = 1/ab
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Solve (Completing the Square): x² - 12x = 5
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x² - 12x + 36 = 5 + 36
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(x - 6)² = 41
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x - 6 = ±√41
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x = 6 ±√41
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Description
Test your skills in algebra with this engaging quiz that covers simplification, solving equations, and factorization. Find your way through various problems, including radical expressions and quadratic equations. Perfect for students looking to improve their algebraic understanding and problem-solving abilities.
