- Rewrite the equation in the standard form \(7x^2 - x - 7 = 0\).
- Identify the coefficients \(a = 7\), \(b = -1\), and \(c = -7\).
- Calculate the discriminant \(\Delta = (-1)^2 - 4(7)(-7) = 197\).
- Solve for \(x\) using the quadratic formula to get \(x = \frac{1 \pm \sqrt{197}}{14}\), which matches option (a).
- The final answer is \(\boxed{a}\).
Explanation
- Writing the Equation in Standard Form
The given quadratic equation is \(7x^2 - x = 7\). To solve it using the quadratic formula, we must first express the equation in the standard form \(ax^2 + bx + c = 0\). Subtracting 7 from both sides, we obtain:
\[7x^2 - x - 7 = 0\]
From this standard form, we can identify the coefficients: \(a = 7\), \(b = -1\), and \(c = -7\).
- Applying the Quadratic Formula
The quadratic formula is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
First, we calculate the discriminant, denoted as \(\Delta = b^2 - 4ac\):
\[\Delta = (-1)^2 - 4(7)(-7)\]
\[\Delta = 1 + 196 = 197\]
With the discriminant determined, we substitute the values of \(a\), \(b\), and \(\Delta\) into the formula:
\[x = \frac{-(-1) \pm \sqrt{197}}{2(7)}\]
\[x = \frac{1 \pm \sqrt{197}}{14}\]
- Conclusion
Comparing our result \(x = \frac{1 \pm \sqrt{197}}{14}\) with the given options, we see that it corresponds to option (a). Thus, the values of \(x\) are \(\frac{1 \pm \sqrt{197}}{14}\).
Examples
The quadratic formula is a fundamental tool for finding the roots of equations in physics and engineering. For example, if you are calculating the trajectory of a projectile or determining the optimal point in an electrical circuit, you will often encounter quadratic equations. Knowing how to systematically apply the quadratic formula allows you to find these critical points accurately, ensuring that your models reflect physical reality.