Problem 14 - Precalculus : Building Concepts and Connections

Question

Solve the quadratic equation by factoring. $$-5 x^{2}+45=0$$

Step-by-Step Solution

Answer

The solutions to the quadratic equation are \(x = 3\) and \(x = -3\)

Step 1: Rewrite the equation

Rewrite the equation to represent the quadratic term properly. By dividing both sides by -5 we get: \(x^2 - 9 = 0\)

Step 2: Factor the quadratic

Factor the equation. The equation is now a difference of two squares, which can be factored as \((a - b)(a + b)\). So, this gives us: \((x - 3)(x + 3) = 0\)

Step 3: Apply the Zero-Product Property

Applying the zero-product property (if \(ab = 0\), then \(a = 0\) or \(b = 0\)), set each factor equal to zero and solve for \(x\). This gives us: \(x - 3 = 0 => x = 3\) and \(x + 3 = 0 => x = -3\)

Key Concepts

Factoring

Factoring is a crucial step in solving quadratic equations. It involves expressing the quadratic expression as a product of two binomials. This method helps convert a complicated polynomial into a simpler structure, making further steps easier. For example, with the quadratic equation \(-5x^2 + 45 = 0\), rewriting it as \(x^2 - 9 = 0\) makes it more straightforward to factor.

• Factoring simplifies solving equations.
• It breaks down complex equations into manageable parts.
• Improves understanding of quadratic forms and solutions.

In the above exercise, notice how recognizing patterns (like the difference of squares, which we will discuss) leads directly to the factors: \((x - 3)\) and \((x + 3)\). This approach is rooted in identifying and manipulating algebraic expressions based on specific, familiar patterns.

Difference of Squares

The 'difference of squares' is a special factoring pattern used frequently in algebra. This technique applies to expressions where one perfect square is subtracted from another. The general form for this pattern is \(a^2 - b^2 = (a-b)(a+b)\).
This predictable pattern greatly simplifies the factoring process.

• Only works when both terms are perfect squares.
• Expresses the subtraction of squares as a product of sums and differences.
• Efficiently factors quadratic expressions like \(x^2 - 9\).

In our example, \(x^2 - 9\) can be factored as \((x - 3)(x + 3)\). Recognizing and applying the difference of squares pattern is vital for efficiently finding factors, especially in quadratic equations.

Zero-Product Property

The zero-product property is a powerful concept that simplifies solving factored equations. It states that if the product of two numbers is zero, then at least one of the numbers must be zero. For instance, if \((x - 3)(x + 3) = 0\), then either \(x - 3 = 0\) or \(x + 3 = 0\).
This property is extremely useful for solving quadratic equations, as it allows each factor to be set equal to zero to solve for the variable.

• Transforms factoring into a tool for finding roots.
• Breaks complex equations into simpler, solvable parts.
• Foundation for many algebraic problem-solving methods.

In the exercise, applying the zero-product property gives us the solutions \(x = 3\) and \(x = -3\). This demonstrates the significant utility of this property in quickly reaching solutions after factoring the original equation.