Problem 33 - Algebra 1

Question

Use the quadratic formula to solve the equation. $$y^{2}+11 y+10=0$$

Step-by-Step Solution

Answer

The solutions to the equation \(y^{2}+11y+10=0\) are \(y = -1\) and \(y = -10\).

Step 1: Identification of coefficients

Identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation. In this case, \(a=1\), \(b=11\), and \(c=10\).

Step 2: Application of Quadratic Formula

Apply the quadratic formula \(y = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\). Substituting the identified values of \(a\), \(b\), and \(c\) into the formulas gives us: \(y = \frac{-11 \pm \sqrt{11^{2} - 4*1*10}}{2*1}\). This simplifies to \(y = \frac{-11 \pm \sqrt{121 - 40}}{2}\), which leads to \(y = \frac{-11 \pm \sqrt{81}}{2}\). Thus, the formula simplifies to \(y = \frac{-11 \pm 9}{2}\).

Step 3: Calculate the roots

Finally, we calculate the two possible values of \(y\). For \(y = \frac{-11 + 9}{2}\), the answer is -1. Similarly, for \(y = \frac{-11 - 9}{2}\), the answer is -10. Therefore, the solutions to the quadratic equation are \(y = -1\) and \(y = -10\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are equations that can be written in the standard form \(ax^2 + bx + c = 0\). It provides a straightforward method to find the roots of any quadratic equation. The formula itself is given by:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

To use this formula, we first need to identify the coefficients \(a\), \(b\), and \(c\) from the equation. The \(\pm\) symbol means there are usually two solutions — two roots. These roots can be real or complex numbers, depending on the discriminant \(b^2 - 4ac\).
When the discriminant is positive, we have two distinct real roots. If it's zero, there's exactly one real root, and if it's negative, the roots are complex numbers. This helps us see not only the solutions but also the nature of the parabola created by the quadratic equation.

Solving Equations

When solving equations, especially quadratic ones, the quadratic formula comes in handy for equations difficult to factor or when other methods are cumbersome. Solving a quadratic equation involves several steps:

• Identifying coefficients: This allows us to plug values into the quadratic formula.
• Simplifying under the square root: Calculate the discriminant \(b^2 - 4ac\). This will determine the next steps.
• Evaluating the square root and simplify: Follow through the calculation carefully to find the roots.

For the given equation \(y^2 + 11y + 10 = 0\), plugging the coefficients \(a=1\), \(b=11\), \(c=10\) into the formula helps us derive the roots accurately. Solving accurately requires attention to detail, as arithmetic mistakes can lead to incorrect answers.

Algebra Concepts

Algebra is about finding unknown variables through equations and expressions, and quadratic equations are a fundamental part of this. Knowing how to manipulate algebraic expressions and equations is key. Here are some core concepts in solving such problems:

• Standard Form: Always rewrite equations in the standard form \(ax^2 + bx + c = 0\).
• Coefficient Identification: These constants in front of the variables are crucial for plugging into formulas accurately.
• Discriminant: Understanding this helps predict the nature of roots — whether they are real or complex.

The quadratic formula is just one technique among several in the algebra toolkit. Other methods, such as completing the square or graphing, also reveal solutions. Mastering these methods and understanding their interconnections strengthen overall algebraic understanding.