To determine which equation shows the quadratic formula used correctly for solving the equation [tex]\(7x^2 = 9 + x\)[/tex], let's first rewrite the equation in standard quadratic form [tex]\(ax^2 + bx + c = 0\)[/tex].
Rearranging [tex]\(7x^2 = 9 + x\)[/tex]:
[tex]\[ 7x^2 - x - 9 = 0 \][/tex]
So we have:
- [tex]\( a = 7 \)[/tex]
- [tex]\( b = -1 \)[/tex]
- [tex]\( c = -9 \)[/tex]
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the quadratic formula:
[tex]\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(7)(-9)}}{2(7)} \][/tex]
[tex]\[ x = \frac{1 \pm \sqrt{1 + 252}}{14} \][/tex]
[tex]\[ x = \frac{1 \pm \sqrt{253}}{14} \][/tex]
Now, let's compare this with the given options:
1. [tex]\( x = \frac{-1 \pm \sqrt{(1)^2-4(7)(9)}}{2(7)} \)[/tex]
- This simplifies to [tex]\( x = \frac{-1 \pm \sqrt{1 - 252}}{14} \)[/tex], which is incorrect.
2. [tex]\( x = \frac{1 \pm \sqrt{(-1)^2-4(7)(9)}}{2(7)} \)[/tex]
- This simplifies to [tex]\( x = \frac{1 \pm \sqrt{1 - 252}}{14} \)[/tex], which is incorrect.
3. [tex]\( x = \frac{-1 \pm \sqrt{(-1)^2+4(7)(9)}}{2(7)} \)[/tex]
- This simplifies to [tex]\( x = \frac{-1 \pm \sqrt{1 + 252}}{14} \)[/tex], which is incorrect.
4. [tex]\( x = \frac{1 \pm \sqrt{(-1)^2+4(7)(9)}}{2(7)} \)[/tex]
- This simplifies to [tex]\( x = \frac{1 \pm \sqrt{1 + 252}}{14} \)[/tex], which is correct.
Therefore, the correct equation that shows the quadratic formula used correctly to solve [tex]\(7x^2 = 9 + x\)[/tex] for [tex]\(x\)[/tex] is:
[tex]\[ x = \frac{1 \pm \sqrt{(-1)^2 + 4(7)(9)}}{2(7)} \][/tex]
