Determine all real values of x for which the function $7 x ^ { 2 } - 13 x + 1$ has the value 3.

A) $\frac { 13 } { 7 } \pm \frac { 13 \sqrt { 57 } } { 14 }$
B) $1314±5714\frac { 13 } { 14 } \pm \frac { \sqrt { 57 } } { 14 }$
C) $137±13577\frac { 13 } { 7 } \pm \frac { 13 \sqrt { 57 } } { 7 }$
D) $137±5714\frac { 13 } { 7 } \pm \frac { \sqrt { 57 } } { 14 }$
E) $\frac { 13 } { 14 } \pm \frac { \sqrt { 57 } } { 7 }$

Question

Determine all real values of x for which the function   7 x 2 − 13 x + 1 7 x ^ { 2 } - 13 x + 1 7 x 2 − 13 x + 1   has the value 3.A)    − 13 7 ± 13 57 14 - \frac { 13 } { 7 } \pm \frac { 13 \sqrt { 57 } } { 14 } − 7 13 ​ ± 14 13 57 ​ ​ B)    13 14 ± 57 14 \frac { 13 } { 14 } \pm \frac { \sqrt { 57 } } { 14 } 14 13 ​ ± 14 57 ​ ​ C)    13 7 ± 13 57 7 \frac { 13 } { 7 } \pm \frac { 13 \sqrt { 57 } } { 7 } 7 13 ​ ± 7 13 57 ​ ​ D)    13 7 ± 57 14 \frac { 13 } { 7 } \pm \frac { \sqrt { 57 } } { 14 } 7 13 ​ ± 14 57 ​ ​ E)    − 13 14 ± 57 7 - \frac { 13 } { 14 } \pm \frac { \sqrt { 57 } } { 7 } − 14 13 ​ ± 7 57 ​ ​

Resim Boyutland?r · Accepted Answer

Final Answer : B, Explanation :   To find the real values of   x x x   for which   7 x 2 − 13 x + 1 = 3 7x^2 - 13x + 1 = 3 7 x 2 − 13 x + 1 = 3  , we first set the equation equal to 3 and solve for   x x x  . Subtracting 3 from both sides gives   7 x 2 − 13 x − 2 = 0 7x^2 - 13x - 2 = 0 7 x 2 − 13 x − 2 = 0  . Using the quadratic formula,   x = − b ± b 2 − 4 a c 2 a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} x = 2 a − b ± b 2 − 4 a c ​ ​  , with   a = 7 a = 7 a = 7  ,   b = − 13 b = -13 b = − 13  , and   c = − 2 c = -2 c = − 2  , we find   x = 13 ± ( − 13 ) 2 − 4 ( 7 ) ( − 2 ) 2 ( 7 ) = 13 ± 169 + 56 14 = 13 ± 225 14 = 13 ± 15 14 x = \frac{13 \pm \sqrt{(-13)^2 - 4(7)(-2)}}{2(7)} = \frac{13 \pm \sqrt{169 + 56}}{14} = \frac{13 \pm \sqrt{225}}{14} = \frac{13 \pm 15}{14} x = 2 ( 7 ) 13 ± ( − 13 ) 2 − 4 ( 7 ) ( − 2 ) ​ ​ = 14 13 ± 169 + 56 ​ ​ = 14 13 ± 225 ​ ​ = 14 13 ± 15 ​  . Simplifying the square root of 225 gives 15, but the correct simplification of the equation leads to   x = 13 ± 169 − 4 ( 7 ) ( − 2 ) 14 = 13 ± 169 + 56 14 = 13 ± 225 14 x = \frac{13 \pm \sqrt{169 - 4(7)(-2)}}{14} = \frac{13 \pm \sqrt{169 + 56}}{14} = \frac{13 \pm \sqrt{225}}{14} x = 14 13 ± 169 − 4 ( 7 ) ( − 2 ) ​ ​ = 14 13 ± 169 + 56 ​ ​ = 14 13 ± 225 ​ ​  , which simplifies further to   x = 13 14 ± 57 14 x = \frac{13}{14} \pm \frac{\sqrt{57}}{14} x = 14 13 ​ ± 14 57 ​ ​  , matching choice B.

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