Question

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

Explanation:

Step1: Rewrite the equation in standard form

First, we need to rewrite the given equation \(7x^{2}=9 + x\) into the standard quadratic form \(ax^{2}+bx + c = 0\). Subtract \(9\) and \(x\) from both sides:
\(7x^{2}-x - 9=0\)
Here, \(a = 7\), \(b=- 1\), and \(c=-9\).

Step2: Recall the quadratic formula

The quadratic formula is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).

Step3: Substitute \(a\), \(b\), and \(c\) into the formula

Substitute \(a = 7\), \(b=-1\), and \(c = - 9\) into the quadratic formula:
\(x=\frac{-(-1)\pm\sqrt{(-1)^{2}-4(7)(-9)}}{2(7)}\)
Simplify \(-(-1)=1\) and \(-4(7)(-9)=4(7)(9)\), so we get:
\(x=\frac{1\pm\sqrt{(-1)^{2}+4(7)(9)}}{2(7)}\)

Answer:

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