Problem 72 - Algebra 2

Question

Solve each equation by factoring. \(x^{2}-5 x=14\)

Step-by-Step Solution

Answer

The solutions are \(x = 7\) and \(x = -2\).

Step 1: Move all terms to one side

First, rewrite the equation so that all terms are on one side of the equation, setting it equal to zero. This is the standard form required for factoring quadratic equations. Subtract 14 from both sides: \[ x^2 - 5x - 14 = 0 \]

Step 2: Factor the quadratic equation

Next, factor the quadratic equation \(x^2 - 5x - 14 = 0\). We need to find two numbers that multiply to -14 (the constant term) and add up to -5 (the coefficient of the middle term). Breaking it down: - The factor pair of -14 that adds to -5 is -7 and 2. - The equation factors to: \[ (x - 7)(x + 2) = 0 \]

Step 3: Solve for x

With the factored form \((x - 7)(x + 2) = 0\), solve for \(x\) by setting each factor equal to zero: 1. \(x - 7 = 0\) leads to \(x = 7\).2. \(x + 2 = 0\) leads to \(x = -2\).

Step 4: Verify the solutions

Substitute \(x = 7\) back into the original equation: \[7^2 - 5(7) = 14 \rightarrow 49 - 35 = 14 \] which is true. Substitute \(x = -2\) back into the original equation: \[ (-2)^2 - 5(-2) = 14 \rightarrow 4 + 10 = 14 \], which is also true. So both solutions are verified.

Key Concepts

Solving Equations

Solving equations, particularly quadratic ones, involves finding the values of the variable that make the equation true. This process can appear complex, but breaking it down into simple steps can make it manageable. When you're dealing with linear equations, you might only need a few steps like addition, subtraction, or division to isolate the variable. However, quadratic equations require a few more techniques.

• Start by rewriting the equation in a standard form, setting it equal to zero if it isn't already.
• When the equation is in the right format, look for ways to simplify or decipher it, like factoring or using specific formulae.

Factoring is one of the primary techniques used to solve quadratic equations, enabling you to find the solutions where the equation equals zero. Once factored, each solution is found by setting the factor equal to zero and solving for the variable.
Understanding the fundamentals of solving equations makes tackling more complicated ones a simpler task. It’s like unlocking a puzzle by finding where each piece fits.

Quadratic Equation

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 the variable. In our exercise, the original quadratic equation simplifies to a convenient form to be factored:
\[ x^2 - 5x - 14 = 0 \]Quadratic equations typically have two solutions, as the curve they represent on a graph - called a parabola - can intersect the x-axis at two points.

• These solutions are found through methods like factoring, using the quadratic formula, or completing the square.
• Our example uses factoring, turning the equation into a product of two binomials: \((x - 7)(x + 2) = 0\).

By factoring, we efficiently find the points where the parabola meets the x-axis, known as the roots or zeroes of the equation.

Algebraic Techniques

Algebraic techniques are essential tools in solving quadratic equations. Factoring is one particularly useful method that involves rewriting an equation as a product of simpler expressions. Here's how we can think about it:

• Identify two numbers that multiply to give the product of the quadratic's leading coefficient (formed by \(a\cdot c\) in \[ax^2 + bx + c\]) and add to equal the middle coefficient \(b\).
• With our example \(x^2 - 5x - 14 = 0\), finding numbers \(-7\) and \(2\) solved the equation since their product is -14 and their sum is -5.

Upon finding these numbers, you can express the quadratic equation as a product of two terms set to zero, which reveals the solutions when each factor equals zero.
Other methods like the quadratic formula or completing the square are also invaluable algebraic techniques. Each method has its strengths, depending on the structure and coefficients of the given equation. The ability to recognize which method simplifies the problem most effectively is a key skill in algebra.