which equation shows the quadratic formula used correctly to solve ( 7 x ^ { 2 } = 9 + x ) for ( x )?\n( x =…

which equation shows the quadratic formula used correctly to solve ( 7 x ^ { 2 } = 9 + x ) for ( x )?\n( x = \frac { - 1 pm sqrt { ( 1 ) ^ { 2 } - 4 ( 7 ) ( 9 ) } } { 2 ( 7 ) } )\n( x = \frac { 1 pm sqrt { ( - 1 ) ^ { 2 } - 4 ( 7 ) ( 9 ) } } { 2 ( 7 ) } )\n( x = \frac { - 1 pm sqrt { ( - 1 ) ^ { 2 } + 4 ( 7 ) ( 9 ) } } { 2 ( 7 ) } )\n( x = \frac { 1 pm sqrt { ( - 1 ) ^ { 2 } + 4 ( 7 ) ( 9 ) } } { 2 ( 7 ) } )

which equation shows the quadratic formula used correctly to solve ( 7 x ^ { 2 } = 9 + x ) for ( x )?\n( x = \frac { - 1 pm sqrt { ( 1 ) ^ { 2 } - 4 ( 7 ) ( 9 ) } } { 2 ( 7 ) } )\n( x = \frac { 1 pm sqrt { ( - 1 ) ^ { 2 } - 4 ( 7 ) ( 9 ) } } { 2 ( 7 ) } )\n( x = \frac { - 1 pm sqrt { ( - 1 ) ^ { 2 } + 4 ( 7 ) ( 9 ) } } { 2 ( 7 ) } )\n( x = \frac { 1 pm sqrt { ( - 1 ) ^ { 2 } + 4 ( 7 ) ( 9 ) } } { 2 ( 7 ) } )

Answer

Explanation:

Step1: Write the equation in standard form

The standard form of a quadratic equation is (ax^{2}+bx + c=0). Given (7x^{2}=9 + x), we can rewrite it as (7x^{2}-x - 9=0). So, (a = 7), (b=-1), and (c=-9).

Step2: Recall the quadratic formula

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

Step3: Substitute the values of (a), (b), and (c) into the quadratic formula

Substitute (a = 7), (b=-1), and (c=-9) into the formula: (x=\frac{-(-1)\pm\sqrt{(-1)^{2}-4\times7\times(-9)}}{2\times7}=\frac{1\pm\sqrt{1 + 252}}{14}=\frac{1\pm\sqrt{253}}{14}). Let's check each option:

  • For the first option: The formula has (-b) wrong ((b=-1), so (-b = 1) not (-1)) and (c=-9) (the formula uses (c = 9) in (4ac) part).
  • For the second option: (x=\frac{1\pm\sqrt{(-1)^{2}-4(7)(9)}) is wrong because (c=-9), so it should be (4ac=4\times7\times(- 9)) (the formula uses (c = 9)).
  • For the third option: The discriminant part (b^{2}-4ac) (here (b=-1), (a = 7), (c=-9), so (b^{2}-4ac=(-1)^{2}-4\times7\times(-9)=1 + 252) not (b^{2}+4ac)).
  • For the fourth option: (x=\frac{-(-1)\pm\sqrt{(-1)^{2}-4\times7\times(-9)}}{2\times7}=\frac{1\pm\sqrt{1+252}}{14}). The formula (x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}) with (a = 7), (b=-1), (c=-9) is applied correctly.

Answer:

The first option: (x=\frac{-1\pm\sqrt{(1)^{2}-4(7)(9)}}{2(7)}) is incorrect. The second option: (x=\frac{1\pm\sqrt{(-1)^{2}-4(7)(9)}}{2(7)}) is incorrect. The third option: (x=\frac{-1\pm\sqrt{(-1)^{2}+4(7)(9)}}{2(7)}) is incorrect. The fourth option: (x=\frac{1\pm\sqrt{(-1)^{2}-4(7)(-9)}}{2(7)}) (equivalent to (x=\frac{1\pm\sqrt{(-1)^{2}+4(7)(9)}}{2(7)}) after simplifying (-4(7)(-9)=4(7)(9))) is correct.