Problem 27 - Algebra 2

Question

Solve each equation by factoring. Then graph. \(-2 x^{2}+12 x-16=0\)

Step-by-Step Solution

Answer

The solutions are \(x = 4\) and \(x = 2\).

Step 1: Set the Equation in Standard Quadratic Form

The given equation is already in the standard quadratic form, which is \(-2x^2 + 12x - 16 = 0\). This is written in the form \(ax^2 + bx + c = 0\), where \(a = -2\), \(b = 12\), and \(c = -16\).

Step 2: Factor out the Greatest Common Factor (GCF)

First, factor out the greatest common factor of all terms. Here, the GCF is \(-2\). So, divide each term by \(-2\):\[-2(x^2 - 6x + 8) = 0\].

Step 3: Factor the Quadratic Expression

Now, factor the quadratic expression \(x^2 - 6x + 8\). We search for two numbers that multiply to \(8\) (the constant term) and add up to \(-6\) (the coefficient of \(x\)), these are \(-4\) and \(-2\). Therefore, the factorization is:\[(x - 4)(x - 2)\].

Step 4: Set Each Factor Equal to Zero

After factoring, set each factor to zero to solve for \(x\):1. \(x - 4 = 0 \Rightarrow x = 4\)2. \(x - 2 = 0 \Rightarrow x = 2\)

Step 5: Graph the Solutions

To graph the solutions, plot the points \(x = 4\) and \(x = 2\) on the x-axis. Since the original quadratic equation is downward-opening (due to the negative leading coefficient), the parabola has these intercepts. Draw the parabola passing through \((4,0)\) and \((2,0)\), indicating it opens downward.

Key Concepts

Factoring

Quadratic equations often come in the form of \( ax^2 + bx + c = 0 \).To solve by factoring, first check if a greatest common factor (GCF) exists and factor it out.In the equation \( -2x^2 + 12x - 16 = 0 \), all terms share a factor of \( -2 \).By factoring \( -2 \),you simplify the equation to \(-2(x^2 - 6x + 8) = 0\).
Next, proceed to factor the quadratic expression inside the parentheses.Look for two numbers that multiply to the constant term and add up to the coefficient of \( x \).In \( x^2 - 6x + 8 \),find numbers \( -4 \) and \( -2 \)since they multiply to \( 8 \)and add up to \( -6 \).Thus, this factors to\((x - 4)(x - 2)\). This method utilizes identifying patterns and using basic multiplication skills to break down more complex expressions.

Graphing Quadratic Functions

Graphing provides a visual representation of solutions to quadratic equations.A quadratic function modeled by\( -2x^2 + 12x - 16 = 0 \)forms a parabola on a graph.To plot the graph correctly, identifying key points such as the vertex and x-intercepts is essential.
From the factored form \((x - 4)(x - 2)\),we know that the roots, or x-intercepts, are located at \( x = 4 \)and \( x = 2 \).These are the points where the graph crosses the x-axis.The negative leading coefficient \( -2 \)indicates that this parabola opens downwards, so it looks like an upside-down U. Mark the intercepts, and sketch the parabola through them, ensuring it curves downwards.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of \( x \)that satisfy the equation.Using factoring, you've already found the intercepts, which represent the solutions.To solve\( -2(x^2 - 6x + 8) = 0 \),set each factor to zero: \( x - 4 = 0 \), resulting in \( x = 4 \),and\( x - 2 = 0 \), leading to \( x = 2 \).
These solutions tell you the x-values where the quadratic function reaches zero.They are crucial when graphing because they locate the specific points where the function intersects the x-axis.Solving through factoring is one method that frequently provides clear, exact solutions, especially when the quadratic easily factors into binomials.