Problem 8 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-14: Solve the equation by factoring. $$ 48 x^{2}+12 x-90=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{5}{4}\) and \(x = -\frac{3}{2}\).

Step 1: Identify and simplify the equation

First, look at the equation: \(48x^{2} + 12x - 90 = 0\). Identify a common factor in all terms, which is 6. Factor it out to simplify the equation to \(6(8x^{2} + 2x - 15) = 0\). The constant factor 6 can be ignored in terms of finding the roots.

Step 2: Factor the quadratic expression

Now, focus on factoring \(8x^{2} + 2x - 15\). Look for two numbers that multiply to \(-120\) (from \(8 \times -15\)) and add to 2, which are 12 and -10. Rewrite the middle term using these numbers: \(8x^{2} + 12x - 10x - 15\).

Step 3: Group and factor each pair

Group the terms: \((8x^{2} + 12x) + (-10x - 15)\). Factor out the greatest common factor from each pair: \(4x(2x + 3) - 5(2x + 3)\).

Step 4: Factor the common binomial

See that both groups contain a common binomial factor: \((2x + 3)\). Factor this out to get: \((4x - 5)(2x + 3) = 0\).

Step 5: Solve each equation for x

Set each factor equal to zero: \(4x - 5 = 0\) and \(2x + 3 = 0\). Solve each for \(x\): \(4x - 5 = 0\) implies \(x = \frac{5}{4}\); \(2x + 3 = 0\) implies \(x = -\frac{3}{2}\).

Key Concepts

Solving Quadratic Equations

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable. Solving these equations often involves finding the values of \(x\) that make the equation true, also known as the roots of the equation. One effective method to solve quadratic equations is by factoring. This involves expressing the quadratic equation as a product of two binomial expressions.
There are different ways to solve quadratic equations:

• Factoring: Used when the quadratic can be broken down into simpler components.
• Quadratic Formula: Useful for any quadratic equation, especially when factoring is arduous.
• Completing the Square: Used to derive the quadratic formula and solve certain quadratics.

When a quadratic equation is set to zero, as in our example \(48x^2 + 12x - 90 = 0\), solving it by factoring involves finding two numbers that satisfy specific conditions related to the coefficient of the terms. Once factored, each factor is set equal to zero and solved for \(x\).

Common Factors in Polynomials

Finding common factors in polynomials is an important step in simplifying expressions and solving equations. A common factor is a value that divides each term of the polynomial without leaving a remainder. In our example \(48x^2 + 12x - 90 = 0\), the common factor among 48, 12, and 90 is 6.
Factoring out common factors helps to simplify the expression, reducing the complexity and making further factorization easier. After factoring 6 out, the expression becomes \(6(8x^2 + 2x - 15) = 0\). Note that the factor 6 does not influence the roots of the quadratic, so it can be disregarded in the subsequent steps.
Identifying a common factor typically involves:

• Finding the greatest common divisor (GCD) of the coefficients.
• Factoring out the GCD from each term.

This process not only simplifies the polynomial but also makes identifying further factors more straightforward.

Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as a product of its factors. For quadratic polynomials, this usually means breaking the expression down into binomial factors. In the example given, we first rewrite the equation as \(8x^2 + 2x - 15\) after factoring out the common factor. Our task is then to factor this expression further.
The technique here is to find two numbers that multiply to the product of the first and last coefficient (\(8 \times -15 = -120\)) and add to the middle coefficient (2). These numbers are 12 and -10. We can thus rewrite the original expression as \(8x^2 + 12x - 10x - 15\).
Factorization of polynomials involves:

• Determining the product-sum relationship of coefficients.
• Rewriting the middle term using identified numbers.
• Ultimately breaking the polynomial into factors.

Once rewritten, factorization can proceed through grouping or other methods to extract binomial factors.

Grouping Method for Factoring

The grouping method for factoring is a useful technique for breaking down polynomials that cannot be easily factored by inspection. It involves rearranging the expression so that it can be split into groups that can each be factored separately.
In our example, we start with the expression \(8x^2 + 12x - 10x - 15\). This can be grouped as \((8x^2 + 12x) + (-10x - 15)\). Each group is then factored individually:

• Factor \(4x\) out of \(8x^2 + 12x\) to get \(4x(2x + 3)\).
• Factor \(-5\) out of \(-10x - 15\) to get \(-5(2x + 3)\).

Recognizing the common binomial \((2x + 3)\) in both terms allows us to factor it out completely, leading to the solution \((4x - 5)(2x + 3) = 0\).
This method highlights key aspects:

• Creating groups that share common factors.
• Extracting and factoring out binomial expressions.
• Simplifying expressions for easier solution finding.

By using these steps, solving even complex-looking polynomials becomes manageable and clear.