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What are the solutions to the equation [tex]$(2x - 5)(3x - 1) = 0$[/tex]?

A. [tex]x = -\frac{5}{2}[/tex] or [tex]x = \frac{1}{3}[/tex]

B. [tex]x = \frac{2}{5}[/tex] or [tex]x = 3[/tex]

C. [tex]x = \frac{5}{2}[/tex] or [tex]x = \frac{1}{3}[/tex]

D. [tex]x = 6[/tex] or [tex]x = 1[/tex]


Sagot :

To solve the equation [tex]\((2x - 5)(3x - 1) = 0\)[/tex], we need to use the property that if the product of two factors is zero, then at least one of the factors must be zero. This is known as the zero-product property.

1. First Factor: Solve [tex]\(2x - 5 = 0\)[/tex]

[tex]\[ 2x - 5 = 0 \][/tex]
Add 5 to both sides:
[tex]\[ 2x = 5 \][/tex]
Divide both sides by 2:
[tex]\[ x = \frac{5}{2} \][/tex]
So, one solution is [tex]\(x = \frac{5}{2}\)[/tex].

2. Second Factor: Solve [tex]\(3x - 1 = 0\)[/tex]

[tex]\[ 3x - 1 = 0 \][/tex]
Add 1 to both sides:
[tex]\[ 3x = 1 \][/tex]
Divide both sides by 3:
[tex]\[ x = \frac{1}{3} \][/tex]
So, another solution is [tex]\(x = \frac{1}{3}\)[/tex].

Therefore, the solutions to the equation [tex]\((2x - 5)(3x - 1) = 0\)[/tex] are [tex]\(x = \frac{5}{2}\)[/tex] and [tex]\(x = \frac{1}{3}\)[/tex].

The correct answer is:
[tex]\[ x = \frac{5}{2} \text{ or } x = \frac{1}{3} \][/tex]