91Ó°ÊÓ

Factoring is a crucial method when solving quadratic equations. It involves expressing the equation in a product of simpler expressions. For the given quadratic equation \[ 0 = d^2 - 5d - 66, \]our goal is to find two numbers that multiply to the product of the coefficient of \(d^2\) (which is 1) and the constant term (which is -66). These two numbers must also add to the coefficient of the middle term, \(d\), which is -5.

• First, calculate the product: \(1 imes -66 = -66\).
• Next, find two numbers that multiply to -66 and add to -5. The numbers -11 and 6 meet these conditions \( (-11 imes 6 = -66\text{ and } -11 + 6 = -5)\).

Break the middle term using these numbers, \( -11d \) and \( 6d \,\) into the original equation like this:\[ d^2 - 11d + 6d - 66 = 0. \]Use factoring by grouping by first grouping \(d^2 - 11d\) and \(6d - 66\), then factor out the common factors:\[ d(d - 11) + 6(d - 11) = 0. \]Finally, factor \(d - 11\) out:\[ (d + 6)(d - 11) = 0. \]By setting each product to zero, you find the solution values for \(d\). This method simplifies finding solutions to the quadratic equation.

In cases where factoring is difficult, the quadratic formula offers a reliable alternative. For a quadratic equation in the form \[ ax^2 + bx + c = 0, \]the solutions \(x\) can be found using:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \]This formula derives from completing the square technique and is useful in finding solutions when quadratic equations have complex or irrational roots.

• Calculate the discriminant: \(b^2 - 4ac\). It indicates the nature of the roots:
  • If positive, there are two distinct real roots.
  • If zero, there's exactly one real root.
  • If negative, the roots are complex.
• Substitute the values for \(a, b,\) and \(c\) into the formula. For our exercise, \(a = 1, b = -5, c = -66\).

Although the quadratic formula was not necessary for solving our specific exercise, it's a powerful tool in your algebra toolbox, ensuring you can always solve any quadratic equation, even those that don't factor neatly.

Quadratic equations are a specific type of polynomial equations, where the highest exponent of the variable is two. More generally, a polynomial equation can include variables raised to any integer power. Quadratics themselves fall into the category of second-degree polynomials:\[ ax^2 + bx + c = 0. \]The challenge is usually finding the roots, i.e., the values of the variable that make the equation true.

• The degree of the polynomial determines the possible number of solutions. A quadratic equation can have up to two distinct roots.
• These can often be determined by factoring, using the quadratic formula, or graphically finding the intercepts with the x-axis.
• In the context of our exercise, identifying the quadratic form was the first step in solving the equation for real-world applications.

Understanding polynomials and their properties can greatly simplify the process of solving equations involving these expressions, particularly when applied in various mathematical and real-world contexts. This foundational knowledge offers a glimpse of more complex polynomial manipulations encountered at advanced levels of mathematics.