Problem 31 - Algebra for College Students

Question

Solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. $$15 x^{2}+34 x+15=0$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Equation Type

The given equation is a quadratic equation in the standard form \(ax^2 + bx + c = 0\) with \(a = 15\), \(b = 34\), and \(c = 15\). Our goal is to factor the quadratic expression.

Step 2: Finding Two Numbers

To factor the quadratic equation, we need to find two numbers that multiply to \(a \cdot c = 15 \times 15 = 225\) and add to \(b = 34\). These two numbers are 25 and 9 because \(25 \times 9 = 225\) and \(25 + 9 = 34\).

Step 3: Rewrite the Middle Term

Rewrite the quadratic expression by splitting the middle term using the two numbers found: \(15x^2 + 25x + 9x + 15 = 0\).

Step 4: Factor by Grouping

Group the terms into two pairs and factor each pair: \((15x^2 + 25x) + (9x + 15) = 0\). Factor out the greatest common factor from each group: \(5x(3x + 5) + 3(3x + 5) = 0\).

Step 5: Factor out the Common Binomial

Notice that \(3x + 5\) is common in both groups. Factor it out: \((5x + 3)(3x + 5) = 0\).

Step 6: Solve for the Roots

Set each factor equal to zero and solve for \(x\):1. \(5x + 3 = 0\) - Subtract 3 from both sides: \(5x = -3\) - Divide by 5: \(x = -\frac{3}{5}\)2. \(3x + 5 = 0\) - Subtract 5 from both sides: \(3x = -5\) - Divide by 3: \(x = -\frac{5}{3}\)

Key Concepts

Factoring Quadratics

Factoring quadratics is a crucial method for solving quadratic equations. It involves rewriting the quadratic equation \[ax^2 + bx + c = 0\] as a product of two binomial expressions. This process is akin to reversing the distribution (FOIL) method in algebra. The primary goal here is to express the trinomial in factored form, making it easier to find the roots of the equation.Key steps in factoring include:

• Identifying the quadratic equation in standard form and isolating the coefficients \(a\), \(b\), and \(c\).
• Finding two numbers that multiply to \(a \cdot c\) and add up to \(b\).
• Splitting the middle term and grouping terms in pairs.
• Factoring each group to find a common binomial factor.

By following these steps, the quadratic expression \(15x^2 + 34x + 15\) can be factored as \((5x + 3)(3x + 5) = 0\). Factoring makes solving the quadratic more straightforward by reducing it to simpler linear equations.

Quadratic Equation

Quadratic equations are polynomial equations of degree two, commonly represented in the form:\[ax^2 + bx + c = 0\]where \(x\) represents an unknown variable, and \(a\), \(b\), and \(c\) are constants with \(a eq 0\). The equation is called 'quadratic' because it involves terms up to the square of \(x\).The standard form allows the application of various methods to find the solutions or roots, such as:

• Factoring, which is effective when the quadratic can be easily expressed as a product of two binomials.
• The Quadratic Formula, useful for any quadratic, regardless of its ability to be factored easily.
• Completing the Square, a method to transform the quadratic into a perfect square trinomial.

Each method offers a different approach depending on the specific characteristics of the quadratic equation you are dealing with. In the given exercise, we used factoring—illustrating that it can be a quick path to the solution when applicable.

Roots of Equations

The roots of a quadratic equation are the values of \(x\) that satisfy the equation. These are the solutions to the equation and are often called zeros since they make the equation equal to zero.In mathematical terms, if you have a factored equation like:\[(5x + 3)(3x + 5) = 0\]then according to the Zero Product Property, each factor can be set to zero:

• \(5x + 3 = 0\) — Solving this results in \(x = -\frac{3}{5}\).
• \(3x + 5 = 0\) — Solving this gives \(x = -\frac{5}{3}\).

These values are the roots or solutions of the quadratic equation. In a graphical sense, they represent the points where the parabola, represented by the quadratic equation, intersects the x-axis. Understanding these intersections helps in visualizing and comprehending how quadratics function in both algebra and geometry.