To determine the value of [tex]\( a \)[/tex] in the quadratic equation [tex]\( x^2 - 4x + a = 0 \)[/tex] such that the quadratic forms a perfect square, we need to ensure that the quadratic can be expressed as [tex]\((x - h)^2\)[/tex] for some value of [tex]\( h \)[/tex].
For a quadratic equation [tex]\( ax^2 + bx + c \)[/tex], a perfect square trinomial must satisfy the condition that its discriminant is zero. The discriminant [tex]\( \Delta \)[/tex] of a quadratic equation [tex]\( ax^2 + bx + c \)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
In our specific equation [tex]\( x^2 - 4x + a = 0 \)[/tex], we have:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = -4\)[/tex]
- [tex]\(c = a\)[/tex]
Substituting these values into the discriminant formula:
[tex]\[ \Delta = (-4)^2 - 4 \cdot 1 \cdot a \][/tex]
Simplifying the expression for the discriminant gives us:
[tex]\[ \Delta = 16 - 4a \][/tex]
For the quadratic equation to be a perfect square, the discriminant must be zero:
[tex]\[ 16 - 4a = 0 \][/tex]
Solving for [tex]\( a \)[/tex]:
[tex]\[ 4a = 16 \][/tex]
[tex]\[ a = 4 \][/tex]
Therefore, the value of [tex]\( a \)[/tex] that makes the quadratic equation [tex]\( x^2 - 4x + a = 0 \)[/tex] a perfect square is [tex]\( a = 4 \)[/tex].
Thus, when [tex]\( a = 4 \)[/tex], the quadratic equation [tex]\( x^2 - 4x + 4 \)[/tex] can be written as:
[tex]\[ (x - 2)^2 = 0 \][/tex]
This confirms that the quadratic [tex]\( x^2 - 4x + 4 \)[/tex] is indeed a perfect square. The value of [tex]\( a \)[/tex] required is [tex]\( \boxed{4} \)[/tex].
