Question

choose the equation that represents the solutions of 0 = 0.25x² - 8x.
option 1: $x = \frac{0.25 \pm \sqrt{(0.25)^2 - (4)(1)(-8)}}{2(1)}$
option 2: $x = \frac{-0.25 \pm \sqrt{(0.25)^2 - (4)(1)(-8)}}{2(1)}$
option 3: $x = \frac{8 \pm \sqrt{(-8)^2 - (4)(0.25)(0)}}{2(0.25)}$
option 4: $x = \frac{-8 \pm \sqrt{(-8)^2 - (4)(0.25)(0)}}{2(0.25)}$

Explanation:

Step1: Recall Quadratic Formula

The quadratic formula for a quadratic equation \( ax^{2}+bx + c = 0 \) is \( x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \).

Step2: Identify \( a \), \( b \), \( c \) from the equation

Given the equation \( 0 = 0.25x^{2}-8x \), we can rewrite it as \( 0.25x^{2}-8x + 0=0 \). So, \( a = 0.25 \), \( b=- 8 \), \( c = 0 \).

Step3: Substitute \( a \), \( b \), \( c \) into the quadratic formula

Substituting into \( x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \), we get:
\( x=\frac{-(-8)\pm\sqrt{(-8)^{2}-4(0.25)(0)}}{2(0.25)}=\frac{8\pm\sqrt{(-8)^{2}-4(0.25)(0)}}{2(0.25)} \)
Looking at the options, the third option (the one with \( x=\frac{8\pm\sqrt{(-8)^{2}-(4)(0.25)(0)}}{2(0.25)} \)) matches this substitution. The first two options have incorrect values for \( a \), \( b \) (they use \( a = 1 \) and incorrect \( b \) values), and the fourth option has an incorrect sign for \( b \) (it uses \( - 8 \) instead of \( 8 \) as \( -b \) when \( b=-8 \), \( -b = 8 \)).

Answer:

The equation \( x=\frac{8\pm\sqrt{(-8)^{2}-(4)(0.25)(0)}}{2(0.25)} \) (the third option among the given choices)