Problem 101 - College Algebra

Question

Solve each equation in Exercises \(83-108\) by the method of your choice. $$x^{2}-4 x=7$$

Step-by-Step Solution

Answer

The roots of the equation are \(x = 7\), and \(x = -3\).

Step 1: Rewrite equation

Firstly, rearrange the given equation, so that it equals zero. The equation became \(x^{2}-4x-7 = 0\).

Step 2: Determine the coefficients

Realize that it is a quadratic equation of the form \(ax^{2} + bx + c = 0\). Accordingly, in this equation, \(a = 1\), \(b = -4\), and \(c = -7\).

Step 3: Apply the Quadratic Formula

Now, the roots of the quadratic equation can be found by using this formula \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\). Plug \(a\), \(b\) and \(c\) into the formula and simplify.

Step 4: Find the roots

Input the values into the formula: \(x = \frac{4 \pm \sqrt{(-4)^{2}-4*1*(-7)}}{2*1}\). After simplifying the equation, the roots of the equation came out to be \(x = 4 + \sqrt{16+28} / 2 = 4+\sqrt{44} / 2 = 7\) and \(x = 4 - \sqrt{44} / 2 = -3\).

Key Concepts

Quadratic Formula

The quadratic formula is an essential tool for solving quadratic equations. A quadratic equation is any polynomial equation of degree two, typically written as \( ax^2 + bx + c = 0 \) where \( a \), \( b \), and \( c \) are coefficients, and \( a eq 0 \).
The quadratic formula is derived by completing the square and is represented as:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This powerful formula allows you to find the roots (solutions) of any quadratic equation, even when the equation cannot be factored easily. The expression under the square root sign, \( b^2 - 4ac \), is called the discriminant, and it plays a key role in determining the nature of the roots. If the discriminant is positive, the equation has two distinct real roots, if it's zero, there is exactly one real root, and if negative, the roots are complex.

Solving Equations

Solving equations involves finding the values of variables that satisfy the given mathematical expression. For quadratic equations, one common method used in textbook problems is the quadratic formula, as it can handle any kind of quadratic equation shape and complexity.
To solve a quadratic equation, you first need to ensure that the equation is in standard form, i.e., everything is set to zero as in \( ax^2 + bx + c = 0 \). This allows the quadratic formula to be applied correctly. Once set in this form, the equation can be easily solved using the quadratic formula by substituting the coefficients into the formula.

• Check if the equation is in the correct form.
• Identify the coefficients \( a \), \( b \), and \( c \).
• Plug these numbers into the quadratic formula.
• Simplify to find the roots.

This step-by-step method ensures accurate solving of quadratic equations and is a crucial part of many algebra courses.

Coefficient Identification

Coefficient identification is a crucial step in solving quadratic equations. Each term in a quadratic equation is associated with a coefficient, which is the numerical factor in front of the variables.
For the quadratic equation \( ax^2 + bx + c = 0 \):

• The coefficient \( a \) is in front of the \( x^2 \) term.
• The coefficient \( b \) is in front of the \( x \) term.
• The coefficient \( c \) is the constant or the term without any \( x \).

Identifying these coefficients accurately is necessary for using the quadratic formula to find solutions. If a mistake is made here, it can lead to incorrect results. In our original example, \( x^2 - 4x - 7 = 0 \), you would identify the coefficients as \( a = 1 \), \( b = -4 \), and \( c = -7 \). This identification allows the correct application of the formula to solve for \( x \).

Root Finding

Root finding in the context of quadratic equations involves determining the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). These values are also known as the solutions of the equation.
Using the quadratic formula, once the equation coefficients have been identified, you simply plug them into the formula and simplify:

• Substitute \( a \), \( b \), and \( c \) into \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Solve the equation under the square root, which is the discriminant.
• Determine the possible values of \( x \) using + and - in the \( \pm \) part of the formula.

For instance, in our solved example, the roots were identified as \( x = 7 \) and \( x = -3 \). Root finding through the quadratic formula ensures that all forms of quadratic equations are solvable. It is a reliable method for discovering the solutions regardless of whether the roots are real or complex.