Domanda del libro di testo
In Exercises 1–30, factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication.
x² − 9xy + 14y²
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0. Review of Algebra
Factoring Polynomials
2:57 minutiProblema 24
Domanda del libro di testo
In Exercises 23–34, determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial. 24. x² + 4x
Guida verificata passo dopo passo1
Identify the binomial: \(x^2 + 4x\).
To make it a perfect square trinomial, find the constant to add. Use the formula \(\left(\frac{b}{2}\right)^2\), where \(b\) is the coefficient of \(x\).
Here, \(b = 4\). Calculate \(\left(\frac{4}{2}\right)^2 = 4\).
Add this constant to the binomial: \(x^2 + 4x + 4\).
Factor the trinomial: \((x + 2)^2\).
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Perfect Square Trinomial
A perfect square trinomial is a quadratic expression that can be expressed as the square of a binomial. It takes the form (a + b)² = a² + 2ab + b² or (a - b)² = a² - 2ab + b². Recognizing this structure is essential for transforming a given binomial into a perfect square trinomial.
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Solving Quadratic Equations by Completing the Square
Completing the Square
Completing the square is a method used to convert a quadratic expression into a perfect square trinomial. This involves adding a specific constant to the expression, which is derived from taking half of the coefficient of the linear term, squaring it, and adding it to the original expression. This technique is crucial for solving quadratic equations and simplifying expressions.
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Factoring Quadratics
Factoring quadratics involves rewriting a quadratic expression as a product of its linear factors. For perfect square trinomials, this means expressing the trinomial in the form (a + b)² or (a - b)². Understanding how to factor these expressions is important for simplifying algebraic expressions and solving equations.
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