Problem 44 - Applied Calculus

Question

Solve each equation by factoring or the Quadratic Formula, as appropriate. $$ 3 x^{2}-27=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 3\) and \(x = -3\).

Step 1: Move constant to the other side

First, we need to isolate the quadratic term. Let's move the constant to the other side of the equation by adding 27 to both sides.This gives us:\[3x^2 = 27\]

Step 2: Divide by the coefficient of the quadratic term

Now we'll divide both sides by 3 to solve for \(x^2\).This gives:\[x^2 = 9\]

Step 3: Solve for x by taking the square root

To solve for \(x\), we take the square root of both sides. Remember that taking the square root gives both a positive and negative solution.\[x = \pm 3\]

Step 4: Conclusion

The solution to the equation \(3x^2 - 27 = 0\) is \(x = 3\) or \(x = -3\).

Key Concepts

Factoring

Factoring is a popular method to solve quadratic equations. When we factor, we aim to express the quadratic equation as a product of two binomials. This forms an equation like \((x - a)(x - b) = 0\).
When we have a factored equation, it implies that at least one of the factors must equal zero. This is known as the zero-product property.

• First, ensure the equation is in standard form: \(ax^2 + bx + c = 0\).
• Next, try to rewrite it as a product of binomials: \((x - a)(x - b) = 0\).
• Solve each binomial to find possible values of \(x\).

Factoring is often straightforward when factors are integers, but can be more difficult with complex numbers. In the given exercise, the equation \(3x^2 - 27 = 0\) was easily solved by isolating \(x^2\) due to its simplicity, rather than traditional factoring.

Quadratic Formula

The quadratic formula is a powerful tool for solving any quadratic equation, especially when factoring is complex or impossible. The formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula results from solving the general form \(ax^2 + bx + c = 0\) by completing the square.

• The part \(b^2 - 4ac\) is known as the discriminant. It determines the nature of the roots.
• A positive discriminant means two real and distinct solutions.
• A zero discriminant indicates a single repeated real root.
• A negative discriminant implies complex roots.

It's crucial to apply this formula carefully, especially when roots are not easily visible through factoring. In simpler equations like \(x^2 = 9\), direct solving is more efficient, although the quadratic formula could technically still be used.

Solving Equations

Solving equations, particularly quadratic ones, involves finding the variable values that make the equation true. In our case, to solve \(3x^2 - 27 = 0\), we performed several steps:

• Firstly, isolate the \(x^2\) term by moving the constant to the other side.
• Then, divide by the coefficient in front of \(x^2\) to further simplify.
• Finally, take the square root of both sides, acknowledging that you will have both positive and negative solutions.

By following these steps logically, you can effectively tackle many quadratic equations, whether by factoring, using the quadratic formula, or as in this case, direct computation. The key is to understand the nature of the equation and choose an appropriate method to reach the solution efficiently.