Question

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

Explanation:

Step1: Rewrite to standard form

Rearrange $7x^2 = 9 + x$ to $7x^2 - x - 9 = 0$.

Step2: Identify coefficients

For $ax^2+bx+c=0$, $a=7$, $b=-1$, $c=-9$.

Step3: Substitute into quadratic formula

Quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$.
Substitute values: $x=\frac{-(-1)\pm\sqrt{(-1)^2-4(7)(-9)}}{2(7)}$, which simplifies to $x=\frac{1\pm\sqrt{(-1)^2+4(7)(9)}}{2(7)}$.

Answer:

$\boldsymbol{x=\frac{1\pm\sqrt{(-1)^2+4(7)(9)}}{2(7)}}$ (the fourth option)