Problem 77 - Algebra A Combined Function

Question

Solve each equation. \(x^{2}-8 x=-12\)

Step-by-Step Solution

Answer

The solutions are \(x = 6\) and \(x = 2\).

Step 1: Set the Equation to Zero

To solve the quadratic equation, we first need to move all terms to one side of the equation to set it equal to zero. The given equation is \(x^2 - 8x = -12\). Add 12 to both sides. \[x^2 - 8x + 12 = 0\]

Step 2: Identify Coefficients for Factoring

We now consider the quadratic in the standard form \(ax^2 + bx + c = 0\). Here, \(a=1\), \(b=-8\), and \(c=12\). We look for two numbers that multiply to 12 (the constant term) and add to -8 (the coefficient of \(x\)).

Step 3: Factor the Quadratic

We see that the numbers -6 and -2 multiply to 12 and add to -8. So, we can factor the quadratic as:\[(x-6)(x-2) = 0\]

Step 4: Solve for the Roots

To find the solutions, set each factor equal to zero:\[x - 6 = 0\] or \[x - 2 = 0\]Thus, solving these gives the solutions:\x = 6\ or \x = 2.

Key Concepts

Factoring

Factoring in quadratic equations is a method used to simplify expressions by expressing them as a product of their factors. It's an essential skill for solving quadratic equations because it allows us to break down complex expressions into simpler parts. When given a quadratic equation in the form of \[ ax^2 + bx + c = 0 \] the goal is to find two numbers that multiply to the constant term \( c \) and add up to the coefficient \( b \). This is done to rewrite the quadratic expression into two binomials: \[ (x - p)(x - q) = 0 \] Here are some friendly tips to help with factoring:

• Identify the type of quadratic: standard form usually needs factoring.
• Use trial and error with potential factor pairs of \( c \).
• Check your work by multiplying the binomials to ensure they expand back to the original expression.

For example, in the equation below: \[ x^2 - 8x + 12 = 0 \] We factor the equation into \[ (x-6)(x-2) = 0 \]. This means \(-6\) and \(-2\) are our factor pairs that satisfy the requirements.

Roots of an Equation

The roots of an equation, also known as solutions or zeros, are the values of \(x\) that make the equation equal to zero. Understanding the roots is crucial because they are essentially the answers to the equation. For a quadratic equation expressed as: \[ ax^2 + bx + c = 0 \], the roots can often be found through factoring, or, if factoring is complicated, by other methods like the quadratic formula or completing the square.In our example:

• We factored the quadratic into: \( (x-6)(x-2) = 0 \)
• Setting each factor equal to zero gives: \( x-6=0 \) or \( x-2=0 \).
• The solutions to these equations are \( x = 6 \) and \( x = 2 \).

These are the roots, and they tell us where the parabola described by the quadratic equation crosses the \( x \)-axis.

Solving Equations

Solving quadratic equations involves finding the values of \( x \) that satisfy the equation, bringing it down to a simplified form where all terms are on one side. This process typically involves several steps, as shown in our example.Start by rewriting the equation in standard form: \[ ax^2 + bx + c = 0 \], which means moving all terms to one side of the equation. Once in standard form, solve it by:

• Factoring the quadratic expression, if possible.
• Using methods like the quadratic formula if it doesn't factor easily.

In our specific case: We rearranged the equation to: \[ x^2 - 8x + 12 = 0 \] Solved by factoring and found the roots \( x=6 \) and \( x=2 \). This shows the usefulness of algebra in solving equations, as well as the logical process of identifying correct solutions to satisfy an equation.