Question

Solve the equation by factoring. \(x^{2}+42=13 x\)

Short answer

The solution of the equation are \(x = 6, 7\)

Step-by-step solution

Rearrange the equation to standard quadratic form

To achieve this, all terms must be on one side of the equation. Move \(13x\) to the left side of the equation to get \(x^2 - 13x + 42 = 0\)

Factorize the quadratic equation

Now, factorize the quadratic equation to get \((x-6)(x-7) = 0\)

Equate each factor to zero and solve for x

Equating each factor to zero gives \(x-6=0\) and \(x-7=0\), which when solved yields \(x = 6, 7\)

Key concepts

Factoring Quadratic Equations

When you're presented with a quadratic equation, factoring is a method you can use to find its solutions. A quadratic equation is typically in the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero. Factoring relies on breaking down the quadratic expression into a product of two binomials.

For instance, if you start with the equation \( x^2 + 42 = 13x \), you'll first want to arrange it into the standard quadratic form \( x^2 - 13x + 42 = 0 \). The next step is to factorize \( x^2 - 13x + 42 \) using factor pairs of the constant term (42) that add up to the coefficient of the \( x \) term (-13). Upon finding these factors, you re-write the quadratic equation as a product of two binomials, precisely as \( (x-6)(x-7) = 0 \) in our example.

Factoring may appear tricky at first, but with practice, you can quickly identify factor pairs and rewrite quadratic equations in a way that's ready to solve for \( x \).

Quadratic Formula

For times when factoring proves difficult, you have the quadratic formula at your disposal. The quadratic formula is a foolproof method that will solve any quadratic equation, regardless of how complex it is. The formula is \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \), where \( a \), \( b \), and \( c \) are the coefficients from the equation in the form of \( ax^2 + bx + c = 0 \).

Using the Quadratic Formula

Let's apply the quadratic formula to the original example. If we were to plug our values into the formula, we would have \( a = 1 \), \( b = -13 \), and \( c = 42 \). Executing the calculations gives us the same solutions that we would find by factoring, confirming that \( x = 6 \) and \( x = 7 \).

The quadratic formula is especially useful as it can even find solutions when they're not whole numbers, ensuring you can always solve the quadratic equation at hand.

Zero Product Property

The zero product property is central to solving quadratic equations by factoring. It states that if the product of two factors is zero, then at least one of the factors must be zero. In mathematical terms, if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \) or both.

If we apply the zero product property to our factored equation \( (x-6)(x-7) = 0 \), we can divide the problem into two simpler equations: \( x-6 = 0 \) and \( x-7 = 0 \). Solving these brings us to the solutions for \( x \), which are 6 and 7 respectively.

The zero product property simplifies solving quadratic equations because after factoring, all that remains is to set each binomial equal to zero and solve for the variable. Always remember that this property only applies when one side of the equation is equal to zero.