- Rearrange the equation to \(x^2 - 2x + 20 = 0\).
- Apply the quadratic formula with \(a=1\), \(b=-2\), and \(c=20\).
- Calculate the roots as \(x = \frac{2 \pm \sqrt{-76}}{2}\).
- Simplify to reach the final answer of \(\boxed{1 \pm \sqrt{19}i}\).
Explanation
- Rearranging the equation into standard form
The equation provided is \(x^2 + 20 = 2x\). To use the quadratic formula, we must first write this equation in the standard form \(ax^2 + bx + c = 0\). Subtracting \(2x\) from both sides, we get:
\[x^2 - 2x + 20 = 0\]
Here, we can identify the coefficients: \(a = 1\), \(b = -2\), and \(c = 20\).
- Applying the quadratic formula
The quadratic formula states that for a quadratic equation \(ax^2 + bx + c = 0\), the solutions for \(x\) are given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Substituting the identified values \(a = 1\), \(b = -2\), and \(c = 20\) into the formula:
\[x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(20)}}{2(1)}\]
Simplifying the expression:
\[x = \frac{2 \pm \sqrt{4 - 80}}{2}\]
\[x = \frac{2 \pm \sqrt{-76}}{2}\]
- Simplifying the solutions
We need to simplify the square root of the negative number \(\sqrt{-76}\). Since \(\sqrt{-76} = \sqrt{76} \times i\), and \(76 = 4 \times 19\), we can write:
\[\sqrt{-76} = \sqrt{4 \times 19 \times (-1)} = 2\sqrt{19}i\]
Substituting this back into our expression for \(x\):
\[x = \frac{2 \pm 2\sqrt{19}i}{2}\]
Dividing both terms in the numerator by 2:
\[x = 1 \pm \sqrt{19}i\]
Comparing this result with the given options, we see that it matches option b.
Examples
The quadratic formula is essential in physics and engineering, such as calculating the trajectory of a projectile. If a projectile's height is modeled by a quadratic equation, setting it equal to zero and using the quadratic formula allows us to find the exact moments the object hits the ground or reaches specific elevations. This mathematical tool provides precise results for complex problems where simple factoring might not work, proving its value in real-world modeling and analysis.