Problem 11 - Algebra and Trigonometry

Question

Solve the equation by factoring. $$ 2 y^{2}+7 y+3=0 $$

Step-by-Step Solution

Answer

The solutions are \( y = -\frac{1}{2} \) and \( y = -3 \).

Step 1: Identify the Quadratic Expression

The given equation is in the standard quadratic form: \[ 2y^2 + 7y + 3 = 0 \]Here, the equation is already set equal to zero, which is the required form to apply the factoring method.

Step 2: Factor the Quadratic Expression

To factor the quadratic expression, first identify the coefficients: \[ a = 2, \ b = 7, \text{ and } \ c = 3. \]We need to find two numbers that multiply to \( ac = 2 \times 3 = 6 \) and add up to \( b = 7. \)The numbers 6 and 1 satisfy these conditions because:\[ 6 \,\cdot\, 1 = 6 \]and\[ 6 + 1 = 7. \]Next, rewrite the middle term using 6 and 1:\[ 2y^2 + 6y + 1y + 3 = 0. \]

Step 3: Group Terms and Factor by Grouping

Group the terms in pairs:\[ (2y^2 + 6y) + (1y + 3) = 0. \]Factor out the greatest common factor in each group:\[ 2y(y + 3) + 1(y + 3) = 0. \]Notice that \( y + 3 \) is a common factor:\[ (2y + 1)(y + 3) = 0. \]

Step 4: Solve the Factored Equation

Set each factor equal to zero and solve for \( y \):1. \( 2y + 1 = 0 \)2. \( y + 3 = 0 \)Solving these equations gives:For \( 2y + 1 = 0 \):\[ 2y = -1 \]\[ y = -\frac{1}{2}. \]For \( y + 3 = 0 \):\[ y = -3. \]

Step 5: Check the Solutions

To ensure the solutions are correct, substitute them back into the original equation.For \( y = -\frac{1}{2} \):\[ 2\left(-\frac{1}{2}\right)^2 + 7\left(-\frac{1}{2}\right) + 3 = 0 \] solves correctly.For \( y = -3 \):\[ 2(-3)^2 + 7(-3) + 3 = 0 \] also works.Both solutions satisfy the original equation.

Key Concepts

quadratic formula

The quadratic formula is a powerful tool to solve quadratic equations of the form \( ax^2 + bx + c = 0 \). Unlike factoring, which is often specific to equations that neatly break down into integer solutions, the quadratic formula works for any quadratic equation. It is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula might look intimidating at first glance, but it’s really just a systematic method to find the values of \( x \) that make the equation true. Here's a simplified explanation:

• \( b \pm \sqrt{b^2 - 4ac} \) involves calculating the term inside the square root, known as the discriminant. This tells you whether you'll have real solutions (when \( b^2 - 4ac \geq 0 \)) and the number of solutions.
• \( -b \) flipping the sign of \( b \) is the step that ensures correct positioning on the x-axis.
• \( 2a \) serves as a scaling factor for the final solution points, ensuring they are accurate.

While the quadratic formula is always applicable, when the solutions might become complex numbers, factoring (if applicable) is often a more straightforward approach, especially for simple equations with integer roots.

factored form

Factoring involves rewriting a quadratic equation in a form that makes it easier to find solutions, often making it look like \((x - p)(x - q) = 0\). This is known as the factored form. For the quadratic equation \( 2y^2 + 7y + 3 = 0 \), the factored form was determined to be \((2y + 1)(y + 3) = 0\). Here's a deeper dive into what's happening when we factor:

• First, you start with the standard form \( ax^2 + bx + c = 0 \) and find two numbers that simultaneously satisfy the multiplication to \( ac \) and addition to \( b \). This step ensures the middle term can be split into two terms that can be grouped.
• In our example, we found 6 and 1 to be the numbers that do this, allowing \( 7y \) to be written as \( 6y + 1y \).
• Once rewritten, grouping is used to identify and pull out common factors, leading to paired terms that can each be factored further.

This method is efficient for equations that factor neatly. It is particularly useful in introductory algebra for solving simple equations and understanding the structure of quadratics.

polynomial equations

A quadratic equation like \( 2y^2 + 7y + 3 = 0 \) is a specific type of polynomial equation. Generally, a polynomial equation is any equation that can be formed from sums of powers of a variable. Quadratics are polynomials of degree 2 because they have the term involving \( y^2 \). Here are some key characteristics:

• Degree is the highest power of the variable in the polynomial, and it dictates the shape of the graph and the number of roots to expect.
• Coefficients, like 2, 7, and 3 in this equation, affect the graph's orientation and width. Changing them alters the solutions.
• Polynomial equations can vary in degree and complexity, but methods like factoring, completing the square, and the quadratic formula apply specifically to quadratic polynomials.

Understanding polynomials includes recognizing the different ways they can appear, such as quadratics, cubics (third degree), and more. This forms a foundational aspect of algebra that extends into higher mathematics and applications.