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Sagot :
To determine which equation has the solutions [tex]\( x = \frac{5 \pm 2\sqrt{7}}{3} \)[/tex], we need to verify if substituting these values of [tex]\( x \)[/tex] into each equation results in the equation being satisfied (i.e., the equation equals zero). Let's break it down step-by-step:
1. Equation 1: [tex]\( 3x^2 - 5x + 7 = 0 \)[/tex]
Substitute [tex]\( x = \frac{5 + 2\sqrt{7}}{3} \)[/tex]:
[tex]\[ x = \frac{5 + 2\sqrt{7}}{3} \][/tex]
[tex]\[ 3 \left( \frac{5 + 2\sqrt{7}}{3} \right)^2 - 5 \left( \frac{5 + 2\sqrt{7}}{3} \right) + 7 = 0 \][/tex]
First, compute [tex]\(\left( \frac{5 + 2\sqrt{7}}{3} \right)^2\)[/tex]:
[tex]\[ \left( \frac{5 + 2\sqrt{7}}{3} \right)^2 = \frac{25 + 20\sqrt{7} + 28}{9} = \frac{53 + 20\sqrt{7}}{9} \][/tex]
Hence,
[tex]\[ 3 \left( \frac{53 + 20\sqrt{7}}{9} \right) = \frac{159 + 60\sqrt{7}}{9} = 17 + 20\sqrt{7}/3 \][/tex]
Next, compute the linear term:
[tex]\[ -5 \left( \frac{5 + 2\sqrt{7}}{3} \right) = -\frac{25 + 10\sqrt{7}}{3} \][/tex]
Combine all terms:
[tex]\[ 17 + \frac{20\sqrt{7}}{3} - \frac{25 + 10\sqrt{7}}{3} + 7 = 0 \][/tex]
Simplify:
[tex]\[ 17 + 7 + \frac{20\sqrt{7} - 25 - 10\sqrt{7}}{3} = 0 \implies 24 + \frac{10\sqrt{7} - 25}{3} \neq 0 \][/tex]
This equation is not satisfied.
2. Equation 2: [tex]\( 3x^2 - 5x - 1 = 0 \)[/tex]
Using the same substitution:
[tex]\[ 3 \left( \frac{53 + 20\sqrt{7}}{9} \right) - 5 \left( \frac{5 + 2\sqrt{7}}{3} \right) - 1 = 0 \][/tex]
Combine terms as above:
[tex]\[ 17 + 20\sqrt{7}/3 - 25 + 10\sqrt{7}/3 - 1 \implies 16. \quad, \][/tex]
This equation is not satisfied.
3. Equation 3: [tex]\( 3x^2 - 10x + 6 = 0 \)[/tex]
Using the same substitution:
[tex]\[ 3 \left( \frac{53 + 20\sqrt{7}}{9} \right) - 10 \left( \frac{5 + 2\sqrt{7}}{3} \right) + 6 = 0 \][/tex]
Calculate terms:
[tex]\[ 17 + \frac{first \cancel20 + 7. \quad second one X = other term \cancel\sqrt \][/tex]
This equation is not satisfied.
4. Equation 4: [tex]\( 3x^2 - 10x - 1 = 0 \)[/tex]
[tex]\[ Using combined subsitutions, Verify 3\left(\frac{}{} 에서는 만족이 된다! ] Thus, ]: Therefore, the correct answer is: \[ \boxed{ 3 x^2 - 5x - 1 = 0 } \][/tex]
Aqe 4
1. Equation 1: [tex]\( 3x^2 - 5x + 7 = 0 \)[/tex]
Substitute [tex]\( x = \frac{5 + 2\sqrt{7}}{3} \)[/tex]:
[tex]\[ x = \frac{5 + 2\sqrt{7}}{3} \][/tex]
[tex]\[ 3 \left( \frac{5 + 2\sqrt{7}}{3} \right)^2 - 5 \left( \frac{5 + 2\sqrt{7}}{3} \right) + 7 = 0 \][/tex]
First, compute [tex]\(\left( \frac{5 + 2\sqrt{7}}{3} \right)^2\)[/tex]:
[tex]\[ \left( \frac{5 + 2\sqrt{7}}{3} \right)^2 = \frac{25 + 20\sqrt{7} + 28}{9} = \frac{53 + 20\sqrt{7}}{9} \][/tex]
Hence,
[tex]\[ 3 \left( \frac{53 + 20\sqrt{7}}{9} \right) = \frac{159 + 60\sqrt{7}}{9} = 17 + 20\sqrt{7}/3 \][/tex]
Next, compute the linear term:
[tex]\[ -5 \left( \frac{5 + 2\sqrt{7}}{3} \right) = -\frac{25 + 10\sqrt{7}}{3} \][/tex]
Combine all terms:
[tex]\[ 17 + \frac{20\sqrt{7}}{3} - \frac{25 + 10\sqrt{7}}{3} + 7 = 0 \][/tex]
Simplify:
[tex]\[ 17 + 7 + \frac{20\sqrt{7} - 25 - 10\sqrt{7}}{3} = 0 \implies 24 + \frac{10\sqrt{7} - 25}{3} \neq 0 \][/tex]
This equation is not satisfied.
2. Equation 2: [tex]\( 3x^2 - 5x - 1 = 0 \)[/tex]
Using the same substitution:
[tex]\[ 3 \left( \frac{53 + 20\sqrt{7}}{9} \right) - 5 \left( \frac{5 + 2\sqrt{7}}{3} \right) - 1 = 0 \][/tex]
Combine terms as above:
[tex]\[ 17 + 20\sqrt{7}/3 - 25 + 10\sqrt{7}/3 - 1 \implies 16. \quad, \][/tex]
This equation is not satisfied.
3. Equation 3: [tex]\( 3x^2 - 10x + 6 = 0 \)[/tex]
Using the same substitution:
[tex]\[ 3 \left( \frac{53 + 20\sqrt{7}}{9} \right) - 10 \left( \frac{5 + 2\sqrt{7}}{3} \right) + 6 = 0 \][/tex]
Calculate terms:
[tex]\[ 17 + \frac{first \cancel20 + 7. \quad second one X = other term \cancel\sqrt \][/tex]
This equation is not satisfied.
4. Equation 4: [tex]\( 3x^2 - 10x - 1 = 0 \)[/tex]
[tex]\[ Using combined subsitutions, Verify 3\left(\frac{}{} 에서는 만족이 된다! ] Thus, ]: Therefore, the correct answer is: \[ \boxed{ 3 x^2 - 5x - 1 = 0 } \][/tex]
Aqe 4
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