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Quadratic substitution is a powerful technique used to simplify polynomial equations. Often, when faced with higher-degree equations with even powers, like the one in the exercise, it's useful to transform them into a more familiar form - a quadratic equation. Here's how it works: If you see an equation involving terms such as\( t^4 \) and \(t^2 \), consider making a substitution.

• Let \( x = t^2 \). This changes the original variable \(t\) to a new variable \(x\).
• In our equation, \(t^4 + 3t^2 = 28\) becomes \(x^2 + 3x = 28\).

This substitution helps reduce the complexity of the equation, making it easier to solve. Once simplified, you'll deal with a standard quadratic equation, which is much easier to manage.

The quadratic formula is your go-to tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). Use it when factoring is difficult or impossible. It is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Applying the Quadratic Formula

In the equation \(x^2 + 3x - 28 = 0\), which we found using substitution, we identify:

• \(a = 1\), \(b = 3\), \(c = -28\)

Plug these into the quadratic formula to find the values of \(x\). The formula simplifies the solving process and provides two possible solutions for \(x\), depending on the calculated discriminant. Once you have \(x\), you can use these values to find the original variable \(t\). Remember, not every solution may be valid in the context of your original equation, so always back-substitute to ensure they make sense.

Understanding the discriminant is key to solving quadratic equations as it offers insights into the nature of the roots. The discriminant, found as \( b^2 - 4ac \), determines how many and what kind of roots exist for the quadratic equation.

The Role of the Discriminant

The roots of a quadratic equation depend on the value of the discriminant:

• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root (a repeated root).
• If negative, the roots are complex or imaginary.

In our case, the discriminant is calculated as:\[3^2 - 4 \times 1 \times (-28) = 9 + 112 = 121\]Since 121 is positive, we know we have two real and distinct roots. This confirmed the validity of both \(x = 4\) and \(x = -7\) as potential solutions, although only the positive result fits the scenario when considering \(x = t^2\). Understanding this step allows you to interpret solution sets accurately and confidently.