22.) \( x^{2}-5=\sqrt{3} x \)

To solve the equation \( x^{2}-5=\sqrt{3} x \), start by rearranging it into standard form. Subtract \( \sqrt{3} x \) from both sides to get: \[ x^2 - \sqrt{3}x - 5 = 0 \] Now, you can apply the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a=1, b=-\sqrt{3}, c=-5 \). Plugging in these values, you'll first calculate the discriminant: \[ b^2 - 4ac = (-\sqrt{3})^2 - 4(1)(-5) = 3 + 20 = 23 \] Now substituting back into the quadratic formula: \[ x = \frac{-(-\sqrt{3}) \pm \sqrt{23}}{2(1)} = \frac{\sqrt{3} \pm \sqrt{23}}{2} \] So the solutions are: \[ x = \frac{\sqrt{3} + \sqrt{23}}{2} \quad \text{and} \quad x = \frac{\sqrt{3} - \sqrt{23}}{2} \] These provide the possible values for \( x \) in your original equation which you can verify by substituting back into the original expression. Now, the equation gives you two values, and this is a perfect chance to explore the effects of quadratic equations! Quadratics are not just academic; they model real-world situations like projectile motion. Want to know what happens with a ball thrown into the air? Quadratic equations describe its height over time — it rises, peaks, then falls. Whether it's sports, physics, or even finance, quadratics give you tools to navigate the world! For further exploration, there's a ton of fascinating math literature out there! Consider diving into "The Art of Problem Solving" by Richard Rusczyk to see how quadratics are just the tip of the mathematical iceberg. Or, try "How to Solve It" by George Pólya for delightful problem-solving strategies that can help you tackle equations like this one and so much more! Happy solving!