Problem 94 - College Algebra

Question

Isolate the terms involving \(y\) on the left side of the equation: $$ y^{2}+2 y+12 x-23-0 $$ Then write the equation in an equivalent form by completing the square on the left side.

Step-by-Step Solution

Answer

The equation in its isolated and squared form is \((y+1)^{2} = 24 + 12x\).

Step 1: Isolating terms involving \(y\)

In this step, isolate the terms involving \(y\) on the left side of the equation. Rearrange the equation to move all \(y\)-containing terms to one side: \(y^{2}+2 y = 23+12x\)

Step 2: Completing the Square

In this step, complete the square on the left side. To do this, add \((b/2)^{2}\) to both sides of the equation, where \(b\) is the coefficient of \(y\). Here, \(b\) is 2. So, \((b/2)^{2} = (2/2)^{2} = 1\).So the equation becomes:\(y^{2}+2 y + 1 = 23 +12x +1\)This left side can now be written as a perfect square. That gives us:\((y+1)^{2} = 24 +12x\)

Step 3: Final result

The final equation after isolating \(y\) and completing the square is:\((y+1)^{2} = 24 + 12x\)This is the result.

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation that involve terms up to the second power. They have the standard form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). These types of equations are prevalent in algebra and often appear in various fields such as physics, engineering, and finance.

The unique aspect of quadratic equations is their graph, which is a parabola. This curve can open upwards or downwards, depending on the sign of the \(a\) coefficient. A crucial skill in working with quadratic equations is the ability to manipulate them to find solutions—often the roots or x-intercepts of the parabola.

There are several methods to solve quadratic equations, including factoring, using the quadratic formula, and completing the square. Each method has its advantages depending on the specific equation. Understanding these techniques is key to mastering quadratic equations.

Algebraic Manipulation

Algebraic manipulation refers to the strategic reorganization and simplification of algebraic expressions to uncover desired information or solve equations. In this exercise, isolating terms involving \(y\) and manipulating the equation accordingly is crucial.

• Move terms around: This is the process of rearranging the equation so all terms involving the variable you are solving for are on one side. In our case, moving terms in \(y^2 + 2y + 12x - 23 = 0\) to form \(y^2 + 2y = 23 + 12x\) sets the stage for further manipulation.
• Balancing equations: When manipulating, you must maintain balance by performing the same operation on both sides. Adding, subtracting, multiplying, or dividing should be mirrored to keep equality intact.
• Simplify: Simplification, such as factoring or expanding, helps make equations more manageable. In our example, this means completing the square, which makes the left side of the equation a perfect square trinomial.

Mastering these skills allows learners to handle more complex equations with confidence and ease.

Solving Equations

Solving equations involves finding the value of variables that satisfy the equation. It often involves various strategies of algebraic manipulation.

When dealing with quadratic equations, solving typically aims to find one or two solutions for the variable. With equations like \((y+1)^2 = 24 + 12x\), after completing the square, the next step would be to use algebraic techniques to isolate \(y\) and solve for it.

• Taking square roots: After writing the expression as a perfect square, like \((y+1)^2\), you can take the square root of both sides. Remembering to consider both positive and negative roots as potential solutions.
• Simplify further: Once you have \(y+1 = \pm \sqrt{24 + 12x}\), isolate \(y\) by subtracting 1 from both sides.
• Verify: Plugging your solution back into the original equation can also help confirm its correctness, ensuring no errors were made during manipulation.

Solving equations is about piecing together the steps methodically until the solution is found, ensuring logical consistency throughout.