Solving for the Square Root: A Guide to Quadratic Equations and Square Roots

In mathematics, understanding and solving quadratic equations and square roots is fundamental. This article will walk you through the process of solving an equation involving a square root. We'll use the problem x - 3x1 √5 as an example, showing you step-by-step how to find the solution.

Understanding the Equation

The given problem is: [x - 3x1]1/2 51/2. This is essentially the square root of x - 3x1 equal to the square root of 5. Squaring both sides of the equation simplifies it significantly.

Squaring Both Sides

When we square both sides, we get:

(x - 3x1)2 (51/2)2

Simplifying the right side, we have:

x - 3x1 5

Rearranging the Equation

Now that we have a standard linear equation, we can rearrange it to solve for x and x1.

x - 3x1 5

Adding 3x1 to both sides:

x - 3x1 3x1 5 3x1

This simplifies to:

x 5 3x1

Forming a Quadratic Equation

Substituting this result back into the original equation, we get a quadratic equation:

x2 - 2x - 3 - 5 0

Combining like terms, we get:

x2 - 2x - 8 0

Solving the Quadratic Equation

This is a standard quadratic equation of the form ax2 bx c 0. We can solve it using the quadratic formula:

x [-b ± √(b2 - 4ac)] / 2a

For our equation, a 1, b -2, and c -8. Plugging these values into the quadratic formula:

x [2 ± √((-2)2 - 4(1)(-8)) / 2(1)]

x [2 ± √(4 32) / 2]

x [2 ± √36 / 2]

x [2 ± 6 / 2]

This gives us two solutions:

x (2 6) / 2 8 / 2 4 x (2 - 6) / 2 -4 / 2 -2

Therefore, the solutions are x 4 and x -2.

Conclusion

Solving for the square root in this equation involves squaring both sides, rearranging the equation to form a quadratic equation, and then using the quadratic formula to find the solutions. In the given problem, the non-negative solution is x 4.

Additional Tips

Simplify the equation before proceeding with more complex operations. Be careful when squaring both sides, as it introduces extraneous solutions that may need to be discarded. Use the quadratic formula to solve any resulting quadratic equations. Always check the solutions to ensure they meet the original equation's requirements.

This guide provides a step-by-step approach to solving equations involving square roots and quadratic equations, which is essential for understanding numerous advanced mathematical concepts.