Problem 61 - Algebra A Combined Function

Question

Solve each equation. $$ 12 y=8 y^{2} $$

Step-by-Step Solution

Answer

The solutions are \(y = 0\) and \(y = \frac{3}{2}\).

Step 1: Rewrite the Equation

To begin, we rewrite the equation in standard quadratic form by subtracting \(12y\) from both sides. This gives us \(8y^2 - 12y = 0\).

Step 2: Factor Out the Greatest Common Factor

Next, we factor out the greatest common factor of the equation. Both terms \(8y^2\) and \(-12y\) share a common factor of \(4y\). So, we factor it out: \(4y(2y - 3) = 0\).

Step 3: Solve Each Factor

Set each factor equal to zero and solve for \(y\).First factor:\(4y = 0\) Divide by 4:\(y = 0\)Second factor:\(2y - 3 = 0\)Add 3 to both sides:\(2y = 3\)Divide by 2:\(y = \frac{3}{2}\).

Step 4: Verify the Solutions

Substitute each solution back into the original equation to ensure they satisfy the equation.For \(y = 0\):Left side: \(12(0) = 0\)Right side: \(8(0)^2 = 0\)Both sides equal, so \(y = 0\) is a solution.For \(y = \frac{3}{2}\):Left side: \(12 \left(\frac{3}{2}\right) = 18\)Right side: \(8 \left(\frac{3}{2}\right)^2 = 18\)Both sides equal, confirming \(y = \frac{3}{2}\) is also a solution.

Key Concepts

Factoring

Factoring is a crucial skill when it comes to solving quadratic equations. It involves expressing a polynomial as the product of its factors. In the case of the equation given in the exercise, we start by arranging all terms to one side to form the standard quadratic equation format. This results in the equation: \(8y^2 - 12y = 0\). The goal is to identify and factor out the common elements present in all terms. Factoring simplifies an equation, making it easier to solve. By recognizing common factors, in this case \(4y\), we simplify our original equation to \(4y(2y - 3) = 0\). This transformation allows us to break down the problem into smaller, more manageable parts that we can solve separately. Remember, factoring not only helps in finding solutions but also provides insights into the roots and characteristics of the equation.

Common Factors

Identifying common factors in terms of a quadratic equation is an essential step toward simplifying and solving it. A common factor is essentially a number or expression that evenly divides all terms of the equation without leaving a remainder.In our exercise, the equation \(8y^2 - 12y = 0\) contains terms that share a common factor, which in this case is \(4y\). By extracting this factor, we rewrite the equation as \(4y(2y - 3) = 0\). Finding a common factor not only simplifies the equation but also reduces computational complexity, making it more straightforward to handle. This step reveals more about the equation’s roots and sets the stage for solving each part individually. When practicing, always check for the greatest common factor as a primary step in simplifying quadratic equations.

Solving Equations

Solving quadratic equations involves finding values of the unknown variable that satisfy the equation. After factoring the original equation into \(4y(2y - 3) = 0\), we can utilize the Zero-Product Property. This principle states if a product of two numbers is zero, then at least one of the factors must be zero.We apply this property by setting each factor equal to zero:

• First, \(4y = 0\), which simplifies to \(y = 0\).
• Second, \(2y - 3 = 0\), solving this gives \(y = \frac{3}{2}\).

Solving these simple equations provides us with the roots of the quadratic equation. To ensure these solutions are correct, we substitute them back into the original equation. Both solutions \(y = 0\) and \(y = \frac{3}{2}\) satisfy \(12y = 8y^2\), confirming they are correct. This verification step is critical, confirming the reliability of the solutions derived. Practice with different equations helps solidify understanding and increase confidence in solving quadratic equations through factoring.