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Magazine Chapter 6: Problem 120
Find all values of \(b\) so that \(0.16 x^{2}+b x y+0.25 y^{2}\) is a perfcct- squarc trinomial.
Short Answer
Expert verified
The value of \(b\) is 0.4.
Step by step solution
01
Identify the General Form of a Perfect Square Trinomial
A quadratic expression of the form \( ax^2 + 2hxy + by^2 \) is a perfect square if it can be written as \((mx + ny)^2\). By expanding \((mx + ny)^2 = m^2x^2 + 2mnxy + n^2y^2\), we identify that \(a = m^2\), \(b = n^2\), and \(2h = 2mn\).
02
Match the Coefficients
Compare the given expression \(0.16x^2 + bxy + 0.25y^2\) with the expanded form \(mx^2 + 2mny + n^2y^2\). We have \(0.16 = m^2\), \(0.25 = n^2\), and \(b = 2mn\).
03
Solve for \(m\) and \(n\)
From \(m^2 = 0.16\), solve for \(m\). We get \(m = \sqrt{0.16} = 0.4\). From \(n^2 = 0.25\), solve for \(n\). We get \(n = \sqrt{0.25} = 0.5\).
04
Calculate the Value of \(b\)
Use the relation \(b = 2mn\) to find \(b\). Substitute \(m = 0.4\) and \(n = 0.5\) into this equation. Thus, \(b = 2 \times 0.4 \times 0.5 = 0.4\).
05
Verify the Perfect Square Condition
Verify that \(0.16x^2 + 0.4xy + 0.25y^2\) is a perfect square trinomial by expanding \((0.4x + 0.5y)^2\), which equals \(0.16x^2 + 0.4xy + 0.25y^2\). The expanded form matches the original trinomial, confirming correctness.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Expression
A quadratic expression is a polynomial of degree 2, typically consisting of three terms: a term with a squared variable, a term with two variables multiplied together, and a constant. These expressions can take several forms, but when considering a potential perfect square trinomial, the expression must be in the form:
\(ax^2 + bxy + cy^2\). Here, \(a\), \(b\), and \(c\) are coefficients. When arranged correctly, this quadratic can be represented as the square of a binomial. Recognizing a quadratic expression's potential to become a perfect square trinomial involves examining these coefficients and their interrelations carefully.
\(ax^2 + bxy + cy^2\). Here, \(a\), \(b\), and \(c\) are coefficients. When arranged correctly, this quadratic can be represented as the square of a binomial. Recognizing a quadratic expression's potential to become a perfect square trinomial involves examining these coefficients and their interrelations carefully.
Coefficient Matching
To transform a quadratic expression into a perfect square trinomial, it鈥檚 crucial to match coefficients accurately. The step involves comparing the given quadratic expression
\(ax^2 + bxy + cy^2\) with its potential perfect square form
\((mx + ny)^2\). By expanding the latter, you get
\(m^2x^2 + 2mnxy + n^2y^2\).
Therefore, you need to ensure:
\(ax^2 + bxy + cy^2\) with its potential perfect square form
\((mx + ny)^2\). By expanding the latter, you get
\(m^2x^2 + 2mnxy + n^2y^2\).
Therefore, you need to ensure:
- \(a = m^2\)
- \(b = 2mn\)
- \(c = n^2\)
Algebraic Expansion
Algebraic expansion is the process of expanding an expression such as
\((mx + ny)^2\) into a polynomial form. When you expand this, you perform the multiplication:
\((mx + ny)(mx + ny)\).
It leads to:
\((mx + ny)^2\) into a polynomial form. When you expand this, you perform the multiplication:
\((mx + ny)(mx + ny)\).
It leads to:
- \(m^2x^2\) from \((mx)(mx)\)
- \(2mnxy\) from \((mx)(ny) + (ny)(mx)\)
- \(n^2y^2\) from \((ny)(ny)\)
Square Roots
Understanding square roots is essential when dealing with perfect square trinomials because these roots help determine the coefficients of the binomial square. Given
\(m^2\) and
\(n^2\),
calculate \(m\) and \(n\) by taking the square root of \(m^2\) and
\(n^2\) respectively. For instance:
\(m^2\) and
\(n^2\),
calculate \(m\) and \(n\) by taking the square root of \(m^2\) and
\(n^2\) respectively. For instance:
- If \(m^2 = 0.16\), then \(m = \sqrt{0.16} = 0.4\)
- If \(n^2 = 0.25\), then \(n = \sqrt{0.25} = 0.5\)
