Problem 34 - Algebra 2

Question

Solve each equation by factoring. \(4 x^{2}=-3 x\)

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = -\frac{3}{4}\).

Step 1: Rearrange the Equation

First, rearrange the equation to set it equal to zero: \[ 4x^2 + 3x = 0 \]

Step 2: Factor the Equation

Next, factor the left side of the equation by taking out the greatest common factor, which is \(x\):\[ x(4x + 3) = 0 \]

Step 3: Apply the Zero Product Property

The Zero Product Property states that if a product is equal to zero, at least one of the factors must be zero. Apply this to solve for \(x\):1. \(x = 0\)2. \(4x + 3 = 0\)

Step 4: Solve Each Factor Equation

Solve each equation from the results of Step 3 for \(x\): 1. \(x = 0\) is already solved.2. For \(4x + 3 = 0\), subtract 3 from both sides and then divide by 4: \[ 4x = -3 \x = -\frac{3}{4} \]

Key Concepts

Factoring Quadratics

Factoring quadratics is one of the most common methods used to solve quadratic equations. This process involves expressing the quadratic in terms of its factors, which are simpler expressions. A quadratic equation typically takes the form \(ax^2 + bx + c = 0\), where \(a, b,\) and \(c\) are constants. To factor this expression, you need to find two numbers that multiply to give you \(ac\) and add to give you \(b\).

• Step 1: Ensure the equation is arranged in standard quadratic form with zero on one side.
• Step 2: Identify values of \(a, b,\) and \(c\).
• Step 3: Find pairs of numbers that multiply to \(ac\) and sum to \(b\).
• Step 4: Rewrite and factor by grouping if necessary.

In our example, we started with the equation \(4x^2 = -3x\). Rearranging it gave us \(4x^2 + 3x = 0\), and by factoring out the common factor \(x\), we got \(x(4x + 3) = 0\). This shows how quadratic equations can be simplified using factoring.

Zero Product Property

The Zero Product Property is a helpful rule that makes solving factored equations straightforward. It tells us that if a product of two expressions equals zero, then at least one of the expressions must be zero. This is because zero is the only number that 'neutralizes' multiplication. When solving a quadratic equation by factoring, once you've expressed it as a product of factors equal to zero, you can apply this property to find the solutions.

• Step 1: Ensure the equation is factored and equals zero.
• Step 2: Set each factor equal to zero and solve.

In the example, we had \(x(4x + 3) = 0\). By applying the Zero Product Property, we concluded that either \(x = 0\) or \(4x + 3 = 0\). This simple reasoning process helps us find potential solutions for any factored quadratic expression.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, meaning they feature a term with an \(x^2\). They typically take the standard form \(ax^2 + bx + c = 0\). Solutions to these equations are usually found using methods like factoring, completing the square, or using the quadratic formula. A quadratic equation can have up to two solutions, reflecting the roots where the function intersects the x-axis of a graph.

• Quadratics can often be represented as parabolas on a graph.
• Not all quadratics are easily factorable; some require the quadratic formula.
• Real solutions represent the x-intercepts of the parabola.

For the equation \(4x^2 + 3x = 0\), we identified the factors and solved them directly. Whether through graphing or algebraic strategies, quadratic equations are fundamental and frequently encountered in algebra.

Greatest Common Factor

The greatest common factor (GCF) is the largest integer that divides two or more numbers without leaving a remainder. When solving equations, factoring out the GCF from polynomial expressions simplifies the task and makes finding solutions easier. It reduces a complex expression into simpler factors, revealing the structure necessary for applying further techniques like the Zero Product Property.

• To find the GCF, determine the largest number that divides all coefficients.
• Factor this number out from each term in the equation.

In our equation \(4x^2 + 3x = 0\), we identified \(x\) as the GCF because it appears in both terms. By factoring it out, \(x(4x + 3)\) simplifies the expression and sets us up to apply the Zero Product Property. Simplifying complex expressions through GCF is a powerful tool in your mathematic toolkit.