91Ó°ÊÓ

To solve a quadratic equation like the one given in the exercise, one of the most helpful tools is the quadratic formula. This formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]The quadratic formula allows us to find the roots or solutions of any quadratic equation of the form \[ ax^2 + bx + c = 0 \]- **a** is the coefficient of the term with \( x^2 \).
- **b** is the coefficient of the term with \( x \).
- **c** is the constant term.
To apply this formula, we first identify the coefficients like we did with \( a = 2 \), \( b = -24 \), and \( c = 72 \) in the given equation.
Next, we substitute these values into the quadratic formula and simplify.

The quadratic formula is very versatile and works for any quadratic equation, whether it has one solution, two solutions, or no real solutions. Understanding how to use it effectively is key to mastering quadratic equations.

The discriminant is a critical part of the quadratic formula, though it often goes unnoticed. It is the part of the formula underneath the square root, noted as: \[ b^2 - 4ac \]The discriminant helps us determine the nature of the solutions of the quadratic equation:

• If the discriminant is positive, there are two distinct real solutions.
• If the discriminant is zero, there is exactly one real solution, as seen in our example where \( 576 - 576 = 0 \).
• If the discriminant is negative, there are no real solutions, but two complex solutions.

Knowing the value of the discriminant before even applying the quadratic formula can save time, and give us useful insights into the type of roots we should expect.

The standard form of a quadratic equation is one of the first things we need to identify. It is written as: \[ ax^2 + bx + c = 0 \]For instance, in the exercise, the equation \( 2x^2 - 24x + 72 = 0 \) is already in standard form.- **a** is 2
- **b** is -24
- **c** is 72

Rewriting any quadratic expression in standard form makes it easier to apply methods like the quadratic formula. It allows us to clearly see and extract the coefficients needed.

Achieving the standard form is essential before solving because it provides a clear and organized structure that aligns with the quadratic formula and other solution methods like factoring or completing the square.