Problem 73 - Algebra for College Students

Question

Solve each of the equations. $$3 x^{2}+7 x=0$$

Step-by-Step Solution

Answer

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

Step 1: Factor the Equation

The given equation is a quadratic equation: \[3x^2 + 7x = 0\] To solve it, we first factor out the greatest common factor, which is \(x\). This gives:\[x(3x + 7) = 0\]

Step 2: Solve Each Factor

According to the zero-product property, if the product of two terms is zero, at least one of the terms must be zero. Therefore, set each factor equal to zero: - \(x = 0\) - \(3x + 7 = 0\)Now, solve for \(x\) in each case.

Step 3: Solve \(3x + 7 = 0\)

Solve the equation \(3x + 7 = 0\) by isolating \(x\). First, subtract 7 from both sides:\[3x = -7\]Then, divide both sides by 3:\[x = -\frac{7}{3}\]

Step 4: Combine Solutions

Combine the solutions from the previous steps. The solutions to the equation \(3x^2 + 7x = 0\) are:\[x = 0\] and \[x = -\frac{7}{3}\]

Key Concepts

Factoring

Factoring is a method used to simplify equations and solve them easily. When given a quadratic equation, like \(3x^2 + 7x = 0\), the first step is to factor it. Factoring means finding the common factor or factors (numbers or expressions that divide into each term without a remainder) of the equation. In this case, the common factor is \(x\), which appears in both terms of the equation. So, you can rewrite the equation as \(x(3x + 7) = 0\). By factoring, we break down the equation into simpler, more manageable parts. This setup helps us solve for \(x\) more conveniently in later steps. Without factoring, solving the equation directly would be more complex.

Zero-Product Property

The zero-product property is a handy rule in algebra. It states that if the product of two expressions equals zero, then at least one of the expressions must equal zero. When we applied this property to the equation \(x(3x + 7) = 0\), we set each factor to zero individually: \(x = 0\) or \(3x + 7 = 0\).

By doing this, we effectively split the problem into two more straightforward equations that we can tackle separately. This powerful property helps us resolve many quadratic equations by turning a complicated expression into simpler, linear parts. Remember: wherever you see a product equalling zero, you're looking at an opportunity to break it into its components using this property!

Solving Equations

Once you have factored the equation correctly and applied the zero-product property, you now need to solve the resulting equations. This involves isolating the variable (in this case, \(x\)) to find its values that make the equation true.

For our equation, we have two simpler equations: \(x = 0\) and \(3x + 7 = 0\). The first equation needs no further solving since \(x = 0\) is already isolated. However, for \(3x + 7 = 0\), we perform basic algebraic operations to solve for \(x\).

First, subtract 7 from both sides to get \(3x = -7\). Next, divide both sides by 3 to isolate \(x\), which results in \(x = -\frac{7}{3}\). Checking that these solutions fit back into the original equation ensures we've solved it correctly.

Quadratic Formula

Sometimes, factoring is not possible, or practical, with complex quadratic equations. That's when the quadratic formula comes into play. The formula helps find solutions for any quadratic equation in the form \(ax^2 + bx + c = 0\). It is given by: \[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\]

This formula provides solutions even if the quadratic equation isn't easily factorable. In our specific example with \(3x^2 + 7x = 0\), factoring and the zero-product property were simple and effective. But should you have an equation where neither factoring nor simple inspection works easily, the quadratic formula can be a reliable tool. Testing this formula on our solved equation validates its versatility, as it yields the same solutions: \(x = 0\) and \(x = -\frac{7}{3}\). Understanding and applying the quadratic formula is crucial for solving diverse quadratic equations efficiently.