91Ó°ÊÓ

A quadratic equation is a type of polynomial equation that generally takes the form \(ax^2 + bx + c = 0\). These equations are called "quadratic" because they involve variables, such as \(x\), raised to the power of two. They can appear in various scientific and mathematical applications and usually involve unknown values for which we need to solve. In our original exercise, by making a substitution \(y = \sqrt{x}\), we are transforming our equation into the quadratic form, \(2y^2 - 7y - 30 = 0\). This transformation simplifies problem-solving greatly, turning a complex equation into one that fits the simpler structure of a quadratic equation.

Solving equations involves finding the values of variables that make an equation true. When working with equations, we use various techniques to isolate variables and solve for them. For quadratic equations, like in Step 3 of the solution, we often use the quadratic formula:

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

This formula allows us to solve any quadratic equation by plugging in values for \(a\), \(b\), and \(c\) from the equation format \(ax^2 + bx + c = 0\). In more straightforward scenarios, factoring or completing the square might be other methods employed.
Regardless of the method, the goal remains clear – find the unknown values that satisfy our equation. After using these techniques, always verify that your solutions work by substituting them back into the original equation.

Square root substitution is a clever technique used to simplify equations, especially when dealing with terms involving \(\sqrt{x}\). By letting \(y = \sqrt{x}\), we effectively remove the square root operation, enabling us to work with polynomial terms. This method was employed in our problem where \(\sqrt{x}\) was present.

• First, we identified that substituting \(y = \sqrt{x}\) would help simplify the equation.
• The substitution transformed the original equation to a familiar quadratic form.

Using square root substitution not only simplifies equations but also opens a path to using powerful polynomial-solving techniques like the quadratic formula. Once substituted, the problem becomes a polynomial equation, vastly simplifying our approach and solution.

Extraneous solutions often appear as an unintended consequence of equation manipulation. They are solutions that emerge from the mathematics but do not satisfy the original problem.

• Such solutions occur most commonly when using methods like substitution or squaring.
• In our problem, after substituting back from \(y\) to \(x\), it's crucial to validate solutions by plugging them back into the original equation \(2x - 7\sqrt{x} - 30 = 0\).

Detection and rejection of extraneous solutions safeguard the accuracy of problem-solving. Always check every derived solution against the initial equation to confirm it is valid.