Problem 33 - Algebra 1: Concepts and Skills

Question

Solve the equation by factoring. $$ x^{2}+8 x=65 $$

Step-by-Step Solution

Answer

The solutions for the equation are \(x = -13\) and \(x = 5\).

Step 1: Rearrange the Equation

The first step will be to rearrange the equation in the standard form \(ax^2 + bx + c = 0\). This can be achieved by bringing the constant term on the right-hand side to the left-hand side. Thus, the equation \(x^2 + 8x = 65\) becomes \( x^2 + 8x - 65 = 0\).

Step 2: Factoring the Equation

Then, factor the equation. The equation \(x^2 + 8x - 65 = 0\) can be factored into \((x + 13)(x - 5) = 0\). This is done by finding two numbers that add up to 8 (the coefficient of the x term) and multiply to -65 (the constant).

Step 3: Solve for x

The final step is to solve for \(x\) by setting each factor equal to zero. So, \(x + 13 = 0\) gives \(x = -13\) and \(x - 5 = 0\) gives \(x = 5\).

Key Concepts

Factoring Quadratic Equations

Solving quadratic equations through the method of factoring is a cornerstone in algebra. Factoring involves rewriting the quadratic equation as the product of two simpler expressions, known as binomials. To factor a quadratic equation like \(x^{2}+8x-65=0\), one must find two numbers that, when multiplied, yield the product of the coefficient of \(x^{2}\) (which is 1 in this case) and the constant term (-65), while also adding up to the coefficient of \(x\) (8). This process is somewhat like reverse engineering the quadratic expression from its expanded form.

Finding these two numbers can sometimes be a trial-and-error process, but once identified, they will form the binomial factors. In this exercise, 13 and -5 were found, giving us the factored form as \(x+13)(x-5)=0\). This is a critical step because it simplifies the problem to applying the zero product property to find the equation's roots.

Quadratic Equations in Standard Form

The standard form of a quadratic equation is expressed as \(ax^{2} + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not equal to zero. This form is pivotal for analyzing and solving quadratic equations since it sets a clear structure for employing methods such as factoring, completing the square, or using the quadratic formula.

In the given problem, the equation \(x^{2} + 8x = 65\) is initially not in the standard form due to the constant term being on the right side. Our first step is to transfer it to the left to achieve the standard form, resulting in \(x^{2} + 8x - 65 = 0\). The equation is now ready for factoring, one of the primary techniques used to solve quadratics when in standard form.

Zero Product Property

The zero product property is fundamental in solving quadratic equations after factoring. It states that if the product of two factors is zero, at least one of the factors must be zero. Applying this property to the factored equation \(x+13)(x-5)=0\), we can deduce that either \(x+13=0\) or \(x-5=0\).

This property enables us to split the factored quadratic equation into two simpler linear equations, which can be easily solved. For our example, setting each factor to zero and solving for \(x\) yields the solutions \(x=-13\) and \(x=5\). It's a crucial last step in the process of finding the roots of the quadratic equation, which are the values of \(x\) that satisfy the original equation.