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Cross multiplication is a powerful technique used to solve proportions, which are equations that demonstrate that two ratios are equivalent. Let's break it down further to understand its application. In a proportion like \( \frac{a}{b} = \frac{c}{d} \), cross multiplication involves multiplying across the diagonal.

• First, multiply \( a \) by \( d \), and then \( b \) by \( c \).
• This gives us the equation: \( a \cdot d = b \cdot c \).

The primary purpose of cross multiplication is to eliminate fractions, making it easier to solve for the variable. When you apply this to proportions like \( \frac{b-5}{3} = \frac{2}{b} \), you get the equation \( b(b-5) = 6 \) after cross multiplying.Cross multiplication helps streamline calculations by transforming a proportion into a simple equation, which we can then solve through algebraic manipulation.

A quadratic equation is a type of polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents an unknown. In our example, we derived a quadratic equation from a proportion:\[ b^2 - 5b - 6 = 0 \]Quadratic equations often lead to two solutions because they can be graphed as parabolas, which typically intersect the \( x \)-axis at two points. To solve a quadratic equation, you can use methods such as:

• Factoring
• Using the quadratic formula
• Completing the square

In this exercise, after converting the proportion to a quadratic equation, we used factoring to solve it. Quadratic equations provide a rich area to explore various solution techniques, each offering unique insights and results.

Factoring quadratics is a technique where you express the quadratic polynomial as a product of two binomials. If we revisit our quadratic \( b^2 - 5b - 6 = 0 \), the goal is to break it down into expressions like \((b - m)(b + n) = 0\).

• The numbers \( m \) and \( n \) must multiply to \( c \) (the constant term, here \(-6\)).
• Simultaneously, they must add up to \( -5 \) (the coefficient of the linear term).

In our example, we found that \( -6 \) and \( 1 \) fit these requirements perfectly, leading us to \((b - 6)(b + 1) = 0\).Once the quadratic is factored, set each factor to zero: \( b-6=0 \) gives \( b=6 \), and \( b+1=0 \) gives \( b=-1 \).This factoring process is effective and often the simplest method for solving quadratic equations, provided the quadratic can be factored neatly. It's a handy tool in mathematical problem-solving, laying a foundation for understanding other algebraic techniques.