Problem 72 - Elementary Algebra

Question

Solve \(y^{2}-8 y=-12\).

Step-by-Step Solution

Answer

Question: Solve the quadratic equation \(y^2 - 8y = -12\) and find the values of y. Answer: The solutions to the quadratic equation are y = 2 and y = 6.

Step 1: Rewrite the equation in standard form

First, we need to rewrite the given equation in standard form (i.e., \(ay^2 + by + c = 0\)). To do this, we simply add 12 to both sides of the equation to get: $$y^2 - 8y + 12 = 0$$

Step 2: Identify the coefficients

Now, we see our equation is in the form of \(ay^2 + by + c = 0\). We need to identify the coefficients a, b and c. In our case: a = 1 (since it is \(1y^2\)) b = -8 c = 12

Step 3: Perform the quadratic formula

The quadratic formula is given by: $$y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Plugging in our coefficients for a, b, and c, we get: $$y = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(12)}}{2(1)}$$ Now, let's simplify the formula step by step: $$y = \frac{8 \pm \sqrt{64 - 48}}{2}$$ $$y = \frac{8 \pm \sqrt{16}}{2}$$ Since \(\sqrt{16} = 4\), we now have two possible values for y: 1. Adding 4 to 8: $$y = \frac{8+4}{2} = \frac{12}{2} = 6$$ 2. Subtracting 4 from 8: $$y = \frac{8-4}{2} = \frac{4}{2} = 2$$ Thus, we find the two solutions to the equation are \(y = 2\) and \(y = 6\).

Key Concepts

Standard Form

To solve a quadratic equation, it is important to first write it in the standard form. The standard form of a quadratic equation is expressed as \(ay^2 + by + c = 0\). This means you must have all terms on one side of the equation equal to zero, and the equation arranged in the order of degree, from the highest power to the lowest.
This makes it easier to identify the coefficients \(a\), \(b\), and \(c\), essential for using the quadratic formula later. Breaking it down:

• \(a\) represents the coefficient of \(y^2\).
• \(b\) is the coefficient of \(y\).
• \(c\) is the constant term.

In our exercise, the original problem is \(y^2 - 8y = -12\). To convert this to standard form, add 12 to both sides, resulting in \(y^2 - 8y + 12 = 0\). Now, it's clearly in the required format, ready for the next step, which involves identifying the coefficients: \(a = 1\), \(b = -8\), \(c = 12\).

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of any quadratic equation, especially when it can't be easily factored. It is expressed as:
\(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
This formula might seem complicated at first glance, but if you break it down, it becomes a lot simpler:

• The term \(-b\) involves taking the negative of the \(b\) coefficient.
• The discriminant, \(b^2 - 4ac\), determines the nature of the roots. It sits under the square root (radical sign).
• The "\(\pm\)" indicates that there are generally two solutions: one involving addition and the other involving subtraction.

Once you simplify the square root and the expression, you're left with two possible solutions for \(y\).
In practice, this formula helps you find solutions precisely, without needing to graph or guess.

Solving Quadratic Equations

Solving quadratic equations can be approached in various ways, depending on the specific problem at hand. The quadratic formula is one method, alongside others like factoring and completing the square. In this exercise, we focus on using the quadratic formula because the equation \(y^2 - 8y + 12 = 0\) doesn't factor cleanly.
Let's illustrate its application using the coefficients we identified: \(a = 1\), \(b = -8\), and \(c = 12\).
First, substitute these values into the quadratic formula:
\(y = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(12)}}{2(1)}\).
Break it down step by step:

• Begin with \(-b\). Here, \(-b = 8\).
• Next, compute the discriminant: \((-8)^2 - 4(1)(12) = 64 - 48 = 16\).
• Then find the square root of the discriminant: \(\sqrt{16} = 4\).
• Finally, solve for \(y\) by calculating \(y = \frac{8 \pm 4}{2}\). You'll get two solutions:
  • \(y = 6\)
  • \(y = 2\)

These solutions indicate the points where the parabola intersects the y-axis. Using the quadratic formula provides an exact answer, ensuring accuracy in your calculations.