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Completing the square is a technique used in algebra to solve quadratic equations. It transforms a quadratic equation into a perfect square trinomial, making it easier to solve. Here's a step-by-step guide:
First, start with the general quadratic equation in the form of:
\[ ax^2 + bx + c = 0 \]
1. Move the constant term to the other side of the equation. For our example, \[ x^2 + 10x = 120 \]
2. Add and subtract the square of half the coefficient of x (i.e., \[ \frac{b}{2} \] ). Here, \[ x^2 + 10x + 25 = 120 + 25 \] (because \[ \frac{10}{2} \] is 5 and \[ 5^2 = 25 \] ).
3. Rewrite the left side as a square of a binomial: \[ (x + 5)^2 = 145\]
4. Solve for x by taking the square root of both sides: \[ x + 5 = \pm \sqrt{145} \] and then isolate x to get: \[ x = -5 \pm \sqrt{145} \] .

Completing the square is particularly helpful when the coefficient of x is even, as it makes the arithmetic simpler.

The quadratic formula is a universal method for solving any quadratic equation. Here's how it works step-by-step:
1. Start with the quadratic equation in standard form: \[ ax^2 + bx + c = 0 \]
2. Identify the coefficients a, b, and c. For our problem: \[ a = 1, \ b = 10, \ c = -120 \]
3. Substitute these values into the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] For our example, it becomes:
\[ x = \frac{-10 \pm \sqrt{10^2 - 4 \cdot 1 \cdot (-120)}}{2 \cdot 1} \]
4. Simplify under the square root: \[ x = \frac{-10 \pm \sqrt{460}}{2} \]
5. Break it down further to get: \[ x = \frac{-10 \pm \sqrt{4 \cdot 115}}{2} \]
\[ x = \frac{-10 \pm 2\sqrt{115}}{2} \]
6. Finally, simplify to find: \[ x = -5 \pm \sqrt{115} \]

The quadratic formula is powerful because it always works, regardless of the coefficients or nature of the roots.

Understanding algebra step-by-step is critical in solving quadratic equations. Follow these steps to break down any problem:
1. Always start by simplifying the given equation.
2. Identify the type of equation you are dealing with. In this case, it’s a quadratic equation.
3. Choose the appropriate method to solve it. We discussed two: Completing the Square and the Quadratic Formula.
4. Work through the steps methodically:
• For Completing the Square, make the equation a perfect square trinomial.
• For the Quadratic Formula, follow the formula, simplifying each part step-by-step.
5. Don't skip steps; write each one down. This helps in reducing errors and understanding the process.
6. Double-check your solution by substituting it back into the original equation.

By following these detailed, step-by-step approaches, you build a strong foundation in algebra, which will be invaluable as you encounter more complex problems.