Problem 9 - A Graphical Approach to College Algebra

Question

Answer each question. Which one of the following equations is set up for direct use of the zero- product property? Solve it. A. \(3 x^{2}-17 x-6=0\) B. \((2 x+5)^{2}=7\) C. \(x^{2}+x=12\) D. \((3 x+1)(x-7)=0\)

Step-by-Step Solution

Answer

The equation is D Solutions: \(x = -\frac{1}{3}\) and \(x = 7\).

Step 1: Identify the Proper Form

The zero-product property applies when an equation is set in a format where two expressions are multiplied together, and the product is zero. This property can be directly used in an equation of the form \((a)(b) = 0\).

Step 2: Analyze Each Option

Check each provided option for the zero-product property form: - Option A: Not factored into two expressions, it's a quadratic equation.- Option B: The expression is in squared form and not factored into two separate products set to zero.- Option C: Has to be rearranged into \((x^2 + x - 12 = 0)\) and then factored.- Option D: Presented in the format \((3x+1)(x-7) = 0\). This is already in the zero-product form.

Step 3: Confirm Correct Equation

From the analysis, option D, \((3 x + 1)(x - 7) = 0\), is the correct equation set up for direct use of the zero-product property.

Step 4: Apply Zero-Product Property

According to the zero-product property, if \((a)(b) = 0\), either \(a = 0\) or \(b = 0\). So set each factor to zero:- \(3x + 1 = 0\)- \(x - 7 = 0\)

Step 5: Solve Each Equation

Solve the equations from the factors:1. For \(3x + 1 = 0\): - Subtract 1: \(3x = -1\) - Divide by 3: \(x = -\frac{1}{3}\)2. For \(x - 7 = 0\): - Add 7: \(x = 7\)

Step 6: Present Solution

The solutions for \((3x+1)(x-7) = 0\) are \(x = -\frac{1}{3}\) and \(x = 7\).

Key Concepts

Factoring Quadratic Equations

When dealing with quadratic equations, understanding how to factor them is crucial. Factoring converts a polynomial into a product of two or more factors. These factors are a simpler form of the original expression and can be used to find the roots of the equation.
For example, consider a typical quadratic equation like \(ax^2 + bx + c = 0\). To factor it, look for two numbers that multiply to \(ac\) (the product of the coefficient of \(x^2\) and the constant term) and add up to \(b\) (the coefficient of \(x\)). This method is commonly called the "trial and error" approach:

• If the quadratic can be expressed as \((px + q)(rx + s) = 0\), then finding \(p, q, r,\) and \(s\) such that these conditions hold enables us to factor the equation.
• Rewriting the middle term using the chosen factors allows the equation to be factored completely.

Remember, not every quadratic is easily factored; some require special techniques or numerical methods. Factoring reduces quadratic equations to simpler expressions that make solving them much simpler.

Solving Quadratic Equations

Solving quadratic equations often involves different strategies, and one common method is by using the Zero-Product Property. Once you have successfully factored a quadratic equation, solving it gets much simpler.
Here's how you do it:

• Once you have the equation in a factored form, like \((x - p)(x + q) = 0\), apply the Zero-Product Property.
• This property states that if the product \((a)(b) = 0\), then either \(a = 0\) or \(b = 0\). With our factored equation, set each factor equal to zero independently.
• For the equation \((x - p) = 0\), solving gives \(x = p\), and for \((x + q) = 0\), solving gives \(x = -q\).

Knowing how to efficiently factor and then solve quadratic equations enables you to find the roots, which are the solutions of the equation, more effectively.

Properties of Equality

The Properties of Equality are foundational in solving any algebraic equation, including quadratic equations. These properties ensure that the operations you perform on an equation are valid, allowing us to solve the equation correctly.
Here's a look at some key properties:

• Additive Property of Equality: If \(a = b\), then \(a + c = b + c\). Adding the same quantity to both sides maintains the equality.
• Subtractive Property of Equality: If \(a = b\), then \(a - c = b - c\). Similarly, subtracting the same quantity does not change the equality.
• Multiplicative Property of Equality: If \(a = b\), then \(a \times c = b \times c\), provided \(c eq 0\). Multiplying both sides by the same non-zero number preserves equality.
• Division Property of Equality: If \(a = b\), then \(\frac{a}{c} = \frac{b}{c}\), where \(c eq 0\). Dividing both sides by the same non-zero number keeps the equation balanced.

These properties allow you to isolate variables and simplify equations step by step, making the problem manageable and ensuring that the equations remain equivalent throughout the solving process.