Problem 18 - Precalculus : Building Concepts and Connections

Question

Solve the quadratic equation by factoring. $$9 x^{2}+6 x+1=0$$

Step-by-Step Solution

Answer

The solution to the quadratic equation \(9x^{2}+6x+1=0\) is \(x=-1/3 \).

Step 1: Confirm standard form of quadratic equation

A standard form of the quadratic equation is \(ax^{2}+bx+c=0 \). Here, \(a = 9, b = 6 \) and \(c = 1\). So, it is already in the standard form.

Step 2: Factor the quadratic equation

The quadratic equation \(9x^{2}+6x+1=0 \) can be factored as \((3x+1)^2 = 0 \). The factoring is done by looking for two numbers that multiply to \(a*c = 9*1=9 \), and add up to \(b=6\). These numbers are 3 and 3. Hence, the equation `9x^{2}+6x+1=0` factors to `(3x+1)^2 = 0`.

Step 3: Solve for x

Set each factor equal to zero to solve for \(x\). This gives \(3x+1 = 0 \). Solving for \(x\) gives \(x= -1/3 \).

Key Concepts

Factoring Quadratic Equations

Factoring is a crucial technique for solving quadratic equations, which are expressed in the form of ax^2 + bx + c = 0. The method of factoring involves rewriting the quadratic in a product of two binomials. A common approach is to look for two numbers that not only multiply to ac (product of the coefficients of x^2 and the constant term) but also add up to b (the coefficient of x).
For example, in the equation 9x^2 + 6x + 1 = 0, we look for two numbers that multiply to give ac (9*1=9) and add up to b (6). These numbers are 3 and 3, indicating that (3x+1)(3x+1) or (3x+1)^2 are the binomial factors. We then set the equation equal to zero to isolate x.

By mastering the art of factoring, you can break down seemingly complex quadratic equations into simpler components, making it easier to find the roots of the equation, which are the values of x that satisfy the equation.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is given by ax^2 + bx + c = 0, where a, b, and c are constants, and 'a' is not equal to zero. This is the form best suited for factoring or for using the Quadratic Formula. If you come across a quadratic equation that is not in this form, you may need to rearrange it by performing operations such as adding or subtracting terms on both sides or combining like terms.

In the given equation, 9x^2 + 6x + 1 = 0, it's clear that the equation is already in standard form, where a = 9, b = 6, and c = 1. When the equation is in this form, it becomes more manageable to identify the next steps for solving, whether by factoring, completing the square, or using the Quadratic Formula.

Solve for x

To solve for x implies finding the value of x that makes the equation true. Once a quadratic equation has been factored, as in the previous steps, each factor can be set equal to zero. This is based on the zero product property which states that if a product of two factors equals zero, then at least one of the factors must also be equal to zero.

For the given example, after factoring (3x+1)^2, you derive at (3x+1) = 0. Solving for x, you subtract 1 from both sides and then divide by 3, leading to the solution x = -1/3. It's important while solving for x to carefully perform inverse operations, such as adding or subtracting and multiplying or dividing, to isolate the variable on one side of the equation.