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For what value(s) of [tex]$x$[/tex] is the following equation true: [tex]$4x^2 - 16x + 16 = 0$[/tex]?

A. [tex]-2, +2[/tex]
B. 1, 4
C. [tex]-2[/tex]
D. [tex]+2[/tex]


Sagot :

To determine the value(s) of [tex]\( x \)[/tex] that satisfy the equation [tex]\( 4x^2 - 16x + 16 = 0 \)[/tex], follow these steps:

1. Identify the coefficients: The given quadratic equation is [tex]\( 4x^2 - 16x + 16 = 0 \)[/tex]. Here, the coefficients are [tex]\( a = 4 \)[/tex], [tex]\( b = -16 \)[/tex], and [tex]\( c = 16 \)[/tex].

2. Calculate the discriminant ([tex]\( D \)[/tex]) of the quadratic equation:
[tex]\[ D = b^2 - 4ac \][/tex]
Plugging in the values:
[tex]\[ D = (-16)^2 - 4 \cdot 4 \cdot 16 \][/tex]
[tex]\[ D = 256 - 256 \][/tex]
[tex]\[ D = 0 \][/tex]
Since the discriminant is zero, there is exactly one unique real solution (a repeated root).

3. Use the quadratic formula to find the root(s). The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{D}}{2a} \][/tex]
Given that [tex]\( D = 0 \)[/tex], the formula simplifies to:
[tex]\[ x = \frac{-b}{2a} \][/tex]
Substitute the coefficients [tex]\( a = 4 \)[/tex], [tex]\( b = -16 \)[/tex], and [tex]\( c = 16 \)[/tex]:
[tex]\[ x = \frac{-(-16)}{2 \cdot 4} \][/tex]
[tex]\[ x = \frac{16}{8} \][/tex]
[tex]\[ x = 2 \][/tex]

Thus, the value of [tex]\( x \)[/tex] that satisfies the equation [tex]\( 4x^2 - 16x + 16 = 0 \)[/tex] is [tex]\( x = 2 \)[/tex].

To summarize, the correct value is [tex]\(\boxed{2}\)[/tex].