Problem 18 - Elementary Algebra

Question

For the following problems, solve the equations. $$ y^{2}=-10 y-9 $$

Step-by-Step Solution

Answer

Answer: The solutions for the equation are y = -1 and y = -9.

Step 1: Rewrite as a quadratic equation

Add 10y and 9 to both sides of the equation to get a quadratic equation: $$ y^2 + 10y + 9 = 0 $$

Step 2: Factor or use the quadratic formula

We can factor this quadratic equation as follows: $$ (y + 1)(y + 9) = 0 $$

Step 3: Solve for y

Set each factor equal to zero and solve for y: $$ y + 1 = 0 \quad \Rightarrow \quad y = -1 $$ $$ y + 9 = 0 \quad \Rightarrow \quad y = -9 $$

Step 4: Write the final solution

The solutions to the equation are y = -1 and y = -9.

Key Concepts

Factoring

Factoring is a method used to solve quadratic equations by expressing the quadratic as a product of its factors. For example, consider the equation \( y^2 + 10y + 9 = 0 \). The idea is to break down the equation into simpler terms, which multiply together to give the original quadratic.
To factor this equation:

• Look for two numbers that multiply to the constant term, 9, and add to the linear coefficient, 10.
• These numbers are +1 and +9.
• Rewrite the equation as \((y + 1)(y + 9) = 0\).

Factoring transforms the quadratic into a product of binomials, making it easier to solve the equation. Each factor can be set to zero, leading to potential solutions. Factoring is a straightforward method when the quadratic can be neatly broken into two binomials. However, not all quadratic equations can be factored easily, in which case other methods like completing the square or using the quadratic formula might be necessary.

Quadratic Formula

The quadratic formula is a universal tool for solving quadratic equations, especially when factoring is challenging or impossible. The formula is given by:\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]The terms \(a\), \(b\), and \(c\) come from the quadratic equation in standard form \( ax^2 + bx + c = 0 \). Here's how to use the quadratic formula:

• Identify \(a\), \(b\), \(c\) in your equation. For \(y^2 + 10y + 9 = 0\), they are \(a = 1\), \(b = 10\), and \(c = 9\).
• Plug these values into the formula.
• Perform any calculations under the square root, known as the discriminant: \(b^2 - 4ac\).
• Simplify to find the solutions for \(y\).

This formula works for all quadratic equations but involves more complex calculations compared to factoring. It's especially handy when the equation does not factor neatly.

Roots of Quadratic Equations

The solutions to a quadratic equation like \( y^2 + 10y + 9 = 0 \) are called its roots. They are the values of \(y\) for which the equation holds true, found by setting each factor to zero or using the quadratic formula. In our example, the roots are \(y = -1\) and \(y = -9\), obtained by both factoring and solving \((y + 1)(y + 9) = 0\).
Finding roots is essential because it helps us understand the points where the graph of the quadratic crosses the y-axis. These are the x-intercepts of the parabola represented by the quadratic equation.
Roots can be real or complex, depending on the discriminant \(b^2 - 4ac\). A positive discriminant indicates two real roots. A zero discriminant means one real root (a repeated root), and a negative discriminant results in two complex roots. Understanding these roots provides deep insights into the nature of the quadratic expression and is key to solving quadratic problems.