A quadratic expression is represented as: [tex]ax^2 + bx + c[/tex]. The expressions when the *'s are replaced are:
- [tex]4a^2 + 28a + 49[/tex].
- [tex]36 -24x + 4x^2[/tex].
- [tex]6.25a^2 + 2.5b + \frac 14 b^2[/tex].
- [tex]\frac{1}{100}b^2 + 2bc + 100c^2[/tex]
For a quadratic expression to be a perfect square binomial, the following must be true:
[tex]b^2 = 4ac[/tex]
[tex](a)\ * + 28a + 49[/tex]
By comparison:
[tex]a = *[/tex]
[tex]b = 28[/tex]
[tex]c = 49[/tex]
So, we have:
[tex]b^2 = 4ac[/tex]
[tex]28^2 = 4 \times a \times 49[/tex]
Make a the subject
[tex]a = \frac{28^2}{4 \times 49}[/tex]
[tex]a = 4[/tex]
Hence, the complete expression is:
[tex]4a^2 + 28a + 49[/tex]
[tex](b)\ 36 - 24x + *[/tex]
By comparison:
[tex]a = *[/tex]
[tex]b = -24[/tex]
[tex]c = 36[/tex]
So, we have:
[tex]b^2 = 4ac[/tex]
[tex](-24)^2 = 4 \times a \times 36[/tex]
Make a the subject
[tex]a = \frac{(-24)^2}{4 \times 36}[/tex]
[tex]a = 4[/tex]
Hence, the complete expression is:
[tex]36 -24x + 4x^2[/tex]
[tex](c)\ 6.25a^2 + * + \frac 14 b^2[/tex]
By comparison:
[tex]a = 6.25[/tex]
[tex]b = *[/tex]
[tex]c = \frac 14b^2[/tex]
So, we have:
[tex]b^2 = 4ac[/tex]
[tex]*^2 = 4 \times 6.25 \times \frac 14b^2[/tex]
[tex]*^2 = 6.25 \times b^2[/tex]
Take square roots of both sides
[tex]* = 2.5b[/tex]
Hence, the complete expression is:
[tex]6.25a^2 + 2.5b + \frac 14 b^2[/tex]
[tex](d)\ * + 2bc + 100c^2[/tex]
By comparison
[tex]a = 100[/tex]
[tex][b] = 2b[/tex]
[tex]c= *[/tex]
So, we have:
[tex]b^2 = 4ac[/tex]
[tex](2b)^2 = 4 \times 100 \times c[/tex]
[tex]4b^2 = 400 \times c[/tex]
Divide both sides by 400
[tex]c = \frac 1{100}b^2[/tex]
Hence, the complete expression is:
[tex]\frac{1}{100}b^2 + 2bc + 100c^2[/tex]
Read more about binomial and monomials at:
https://brainly.com/question/2150782
