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Whenever you come across a quadratic equation like \(5x^2 + x - 2 = 0\), your goal is to find the values of \(x\) that make this equation true. This is known as solving the equation.
For quadratic equations, which are polynomial equations of degree two, there is a special method to find the solutions called the Quadratic Formula. The quadratic formula is:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

In the given equation, you start by identifying the values of \(a\), \(b\), and \(c\) in the standard form of the quadratic equation \(ax^2 + bx + c = 0\). Here, \(a = 5\), \(b = 1\), and \(c = -2\). Substitute these values into the quadratic formula to begin solving for \(x\). This method always involves substituting numbers correctly and then simplifying the resulting expression to find the solutions for \(x\).

The roots of a quadratic equation are the values of \(x\) that satisfy the equation, which in simpler terms, are the solutions of the equation. Using the quadratic formula, we can determine these roots straightforwardly.
Once you plug \(a = 5\), \(b = 1\), and \(c = -2\) into

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

you get

• \(x = \frac{-1 \pm \sqrt{1 + 40}}{10}\).
• Simplifying further gives \(x = \frac{-1 \pm \sqrt{41}}{10}\).

Here, the term \(\pm\) indicates two separate operations: addition and subtraction. These determine the two different roots:

• \(x = \frac{-1 + \sqrt{41}}{10}\)
• \(x = \frac{-1 - \sqrt{41}}{10}\).

These roots can be either real or complex numbers, depending on the discriminant \(b^2 - 4ac\). In this case, since \(\sqrt{41}\) is a real number, both roots are real.

Solving math problems, like finding the roots of a quadratic equation, requires a strategic approach. Understanding each part of the problem and how they connect is crucial.

• **Identify:** Start by identifying what's given, such as coefficients \(a\), \(b\), and \(c\) in a quadratic equation.
• **Choose a Method:** Use appropriate methods, like the quadratic formula for second-degree equations.
• **Apply the Method:** Carefully substitute the identified values into the chosen formula.
• **Simplify:** Simplification is key to discerning the true solutions.
• **Verify:** Always verify your solutions by substituting them back into the original equation to check if they hold true.

Breaking down the problem into these smaller steps makes solving complex problems like quadratic equations more manageable. Developing your problem-solving skills enables you to tackle not just equations but all sorts of mathematical challenges effectively.