Question

Solve the quadratic equation using the Quadratic Formula. Then solve the equation using another method. Which method do you prefer? Explain.\(-31 x+56=-x^2\)

Short answer

The Quadratic Formula yields the same solutions as Factoring. Preferences vary, but the Quadratic Formula may be more reliable as it can be used with any quadratic equation.

Step-by-step solution

Set the equation to the standard form

Reorder the equation so that it's in standard form \(ax^2 + bx + c = 0\), then it becomes \(x^2 + 31x - 56 = 0\). Here, \(a = 1\), \(b = 31\), and \(c = -56\)

Use the Quadratic Formula

Apply the Quadratic Formula, which is \(x = {-b \pm \sqrt{b^2 - 4ac} \over 2a}\). Substitute \(a\), \(b\), and \(c\) into the formula to get \(x = {-31 \pm \sqrt{31^2 - 4*1*(-56)} \over 2*1}\). Solve this to find the two solutions for \(x\)

Use Factoring

As a second method, use factoring. First, find two numbers that multiply to -56 (c) and add to 31 (b). These numbers are 32 and -24. So the equation can be rewritten as \(x^2 + 32x - 24x -56 = 0\). Factor by grouping to get \((x - 1)(x + 56) = 0\). Solve for \(x\) in each to find the solutions for \(x\)

Compare the methods

Compare the two methods. The Quadratic Formula is more systematic and works for all quadratic equations, while Factoring is quick and easy but only works when it's easy to find the two numbers

Key concepts

Factoring

Factoring is a method used to simplify complex expressions by expressing them as a product of simpler factors, or terms. When dealing with quadratic equations, factoring becomes an invaluable tool, allowing us to delve into the roots of the equation quickly. A quadratic equation is typically in the form of \( ax^2 + bx + c = 0 \). The goal of factoring is to rewrite this equation in the form \((px + q)(rx + s) = 0\), which allows us to find the values of \(x\) by setting each factor equal to zero.

Here is how we apply this method:

• Determine the values of \(a\), \(b\), and \(c\) from the equation.
• Identify numbers that multiply to \(a \times c\) and add to \(b\).
• Rewrite the middle term using these identified numbers.
• Factor by grouping the terms.

In our exercise, we found numbers 32 and -24 that multiply to -56 and add to 31. By regrouping and factoring, we rewrote the quadratic equation as \((x - 1)(x + 56) = 0\). After factoring, you solve for \(x\) by setting \((x - 1) = 0\) and \((x + 56) = 0\), thus finding the solutions \(x = 1\) and \(x = -56\). Factoring is efficient for equations where the factors are easy to spot, making it a quick way to solve quadratic equations.

Quadratic equations

Quadratic equations are a fundamental component of algebra that deal with polynomials of degree two. The general form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants with \(a eq 0\). Solving quadratic equations involves finding values of \(x\) that satisfy the equality. These values are known as the roots or solutions of the equation. Quadratic equations can often be solved by several methods, including factoring, using the quadratic formula, or graphing.

The behavior and properties of the quadratic equation are often depicted on a graph as a parabola. The parabola can open upwards or downwards depending on the sign of \(a\). When solving these equations, identifying the correct method is key as each method has its own application and efficiency. The quadratic formula, factoring, and completing the square are some of the prominent methods, each suiting different kinds of problems.

Methods of solving equations

There are multiple methods available for solving quadratic equations. Each method provides a unique approach and can serve well depending on the equation at hand. Here, we compare the most commonly used methods: the quadratic formula and factoring.

**The Quadratic Formula**

• Universally applicable for all quadratic equations, ensuring robust and reliable results.
• It follows a straightforward process: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• This method is especially useful when the equation does not easily factor.
• Provides exact solutions, even for complex roots involving imaginary numbers.

**Factoring**

• Provides a quick solution when the equation can easily be expressed as a product of binomials.
• Faster in practice but limited to equations that are easily factorable.
• Often requires some intuition or inspection to identify the factors that work.
• Not applicable when the roots involve irrational or complex numbers.

Choosing a method typically depends on the equation itself and personal preference. While the quadratic formula is a reliable all-purpose tool, factoring can offer speed and simplicity for the right problems.