Answer:
See explanation below
Step-by-step explanation:
What is a perfect square quadratic?
A perfect square quadratic is a quadratic expression that factors into two identical binomials. If a quadratic is a perfect square then the quadratic can be expressed as (ax + b)² where a and b are constants.
There is only one root.
What are the roots of a quadratic equation?
The roots of a quadratic equation can be found using the quadratic formula:
[tex]x = \dfrac{ -b \pm \sqrt{b^2 - 4ac}}{ 2a }[/tex]
The term under the √ sign, b² - 4ac is known as the determinant (D)
In particular when D is zero, we end up with only one root
So for a perfect square quadratic, the condition b² - 4ac applies
Let's compare the given quadratic
[tex](5p-1)x^2 - 4x + (2p-1)[/tex]
with the generalized formula which is
[tex]ax^2 + bx + c[/tex]
Mapping the coefficients of both we get
[tex]a = (5p - 1)\\b = - 4\\c = (2p - 1)\\[/tex]
Let's apply the property for a perfect square formula which is
[tex]b^2 - 4ac = 0[/tex]
[tex](-4)^2 - 4(5p-1)(2p - 1) = 0[/tex]
Using FOIL method,
[tex](5p-1)(2p - 1) = 10p^2 - 7p + 1[/tex]
So
b² - 4ac = 0 becomes
[tex]16 - 4( 10p^2 - 7p + 1) = 0[/tex]
Divide throughout by 4 to get
[tex]4 - (10p^2 - 7p + 1) = 0\\\\4 - 10p^2 + 7p - 1 = 0\\\\-10p^2 + 7p + 3 = 0\\\\[/tex]
Multiply both sides by -1 (not really necessary but makes it easer to understand)
=> [tex]10p^2 - 7p - 3 = 0[/tex]
We can solve using quadratic formula or factorization
Let's factor the above expression as follows:
[tex]10p^2 - 7p - 3 = (10p + 3) (p - 1)[/tex]
Therefore we get
[tex](10p + 3) (p - 1) = 0[/tex]
This means that the two solutions are given by
[tex]10p + 3 = 0\\ = > 10p = - 3\\\\= > p = - \dfrac{3}{10}[/tex]
[tex]p - 1 = 0\\= > p = 1[/tex]
Hence the given quadratic becomes a perfect square when
[tex]p = 1 \;or\;p = -\dfrac{3}{10}[/tex]
