To solve the quadratic equation [tex]\( 7x^2 = 9 + x \)[/tex] for [tex]\( x \)[/tex], we need to first rearrange it into the standard quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex].
Starting from:
[tex]\[ 7x^2 = 9 + x \][/tex]
Rearrange it to:
[tex]\[ 7x^2 - x - 9 = 0 \][/tex]
This gives us the coefficients:
[tex]\[ a = 7, \][/tex]
[tex]\[ b = -1, \][/tex]
[tex]\[ c = -9 \][/tex]
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Plugging in the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
1. Calculate [tex]\( b^2 \)[/tex]:
[tex]\[ (-1)^2 = 1 \][/tex]
2. Calculate [tex]\( 4ac \)[/tex]:
[tex]\[ 4 \cdot 7 \cdot (-9) = -252 \][/tex]
3. Compute the discriminant [tex]\( b^2 - 4ac \)[/tex]:
[tex]\[ 1 - (-252) = 1 + 252 = 253 \][/tex]
So, the quadratic formula for this equation becomes:
[tex]\[ x = \frac{-(-1) \pm \sqrt{253}}{2 \cdot 7} \][/tex]
This simplifies to:
[tex]\[ x = \frac{1 \pm \sqrt{253}}{14} \][/tex]
Therefore, the correct equation showing 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]
Looking at the given options, we see:
[tex]\[ x = \frac{1 \pm \sqrt{(-1)^2 - 4(7)(9)}}{2(7)} \][/tex]
This matches our derived formula, confirming that the correct answer is:
[tex]\[ x=\frac{1 \pm \sqrt{(-1)^2-4(7)(9)}}{2(7)} \][/tex]
