Problem 31 - Algebra A Combined Function

Question

Solve. $$ (x+4)(x-9)=4 x $$

Step-by-Step Solution

Answer

The solutions are \(x = 12\) and \(x = -3\).

Step 1: Expand the Equation

First, expand the left-hand side of the equation \((x+4)(x-9) = x^2 - 9x + 4x - 36\).Combine like terms to obtain:\(x^2 - 5x - 36\).

Step 2: Set Equation to Zero

Bring all terms to one side to set the equation to zero:\(x^2 - 5x - 36 = 4x\).Subtract \(4x\) from both sides:\(x^2 - 5x - 4x - 36 = 0\).Simplify to get:\(x^2 - 9x - 36 = 0\).

Step 3: Factor the Quadratic Equation

Factor the quadratic equation \(x^2 - 9x - 36 = 0\).Look for two numbers that multiply to \(-36\) and add to \(-9\). These numbers are \(-12\) and \(3\).So, the factorization is:\((x - 12)(x + 3) = 0\).

Step 4: Solve for x

Set each factor equal to zero and solve for \(x\):1. \(x - 12 = 0 \Rightarrow x = 12\)2. \(x + 3 = 0 \Rightarrow x = -3\)

Key Concepts

Factoring

Factoring is a method used to simplify expressions or solve equations by breaking up an expression into components, or 'factors,' that when multiplied together return the original expression. For example, to factor the quadratic equation \(x^2 - 9x - 36 = 0\), we seek two numbers whose product is \(-36\) (the constant term) and whose sum is \(-9\) (the coefficient of \(x\)).

• The numbers \(-12\) and \(3\) meet the criteria, since \(-12 \times 3 = -36\) and \(-12 + 3 = -9\).
• Thus, the equation can be rewritten as \((x - 12)(x + 3) = 0\).

This factoring transforms a quadratic equation into a simpler form where we can easily find solutions by setting each factor to zero.

Expanding Expressions

Expanding expressions involves multiplying out a simplified expression to obtain an equivalent but possibly more elaborate form. It is essentially the reverse of factoring. In the given exercise, the expression \((x+4)(x-9)\) is expanded.

• First, distribute \(x + 4\) across \(x - 9\), which results in: \(x(x - 9) + 4(x - 9)\).
• Carrying out the multiplication gives \(x^2 - 9x + 4x - 36\).
• We combined like terms next, resulting in the expression \(x^2 - 5x - 36\).

Expanding can be helpful to simplify or prepare an equation for further operations, such as combining like terms or setting an equation to zero.

Combining Like Terms

Combining like terms is an essential step in simplifying expressions and solving equations. It involves adding or subtracting terms that have the same variable and exponent.

• For instance, when expanding \((x+4)(x-9)\) to \(x^2 - 9x + 4x - 36\), the terms \(-9x\) and \(4x\) are like terms.
• They simplify to \(-5x\) because \(-9x + 4x = -5x\).

This process of combining like terms helps in achieving a simpler expression, making it easier to manipulate or solve.

Solving Equations

Solving equations is the process of finding the value(s) of the variable(s) that satisfy the equation. In quadratic equations, such as \(x^2 - 9x - 36 = 0\), solving often involves factoring the equation, setting each factor to zero, and then solving for the variable.

• Using the factored form \((x - 12)(x + 3) = 0\), each factor is set equal to zero.
• This yields two simple linear equations: \(x - 12 = 0\) and \(x + 3 = 0\).
• Solving these, we find \(x = 12\) or \(x = -3\).

In general, the solutions to a quadratic equation give the points where the expression equals zero, which are also the x-intercepts of the corresponding quadratic function.