Problem 32 - Algebra for College Students

Question

Use the method of completing the square to solve each quadratic equation. $$y^{2}-9 y+30=0$$

Step-by-Step Solution

Answer

The solutions are \( y = \frac{9}{2} \pm \frac{\sqrt{39}i}{2} \).

Step 1: Identify the Quadratic Equation

The given quadratic equation is \( y^2 - 9y + 30 = 0 \). Notice that it is in the form \( ay^2 + by + c = 0 \) where \( a = 1 \), \( b = -9 \), and \( c = 30 \).

Step 2: Move Constant Term to the Right Side

Start by moving the constant term to the right side of the equation: \( y^2 - 9y = -30 \). This will prepare the equation for completing the square.

Step 3: Find the Value to Complete the Square

To complete the square, take half of the coefficient of \( y \) (which is -9), square it, and add it to both sides. Half of -9 is \( \frac{-9}{2} = -\frac{9}{2} \). Squaring it, we get \( \left(-\frac{9}{2}\right)^2 = \frac{81}{4} \).

Step 4: Add and Subtract the Square Value

Add and subtract \( \frac{81}{4} \) inside the equation: \( y^2 - 9y + \frac{81}{4} = -30 + \frac{81}{4} \). Now the left side becomes a perfect square trinomial.

Step 5: Rewrite as a Perfect Square

Rewrite the left side as \( (y - \frac{9}{2})^2 \): \( (y - \frac{9}{2})^2 = -30 + \frac{81}{4} \).

Step 6: Simplify the Right Side

Combine the terms on the right: Convert -30 to a fraction with a denominator of 4: \( -30 = \frac{-120}{4} \). Therefore, \( \frac{-120}{4} + \frac{81}{4} = \frac{-39}{4} \).

Step 7: Solve for y by Taking the Square Root

Take the square root of both sides: \( y - \frac{9}{2} = \pm \sqrt{\frac{-39}{4}} \). Since the square root of a negative number involves \( i \), we have \( y - \frac{9}{2} = \pm \frac{\sqrt{39}i}{2} \).

Step 8: Solve for y

Isolate \( y \) by adding \( \frac{9}{2} \) to both sides: \( y = \frac{9}{2} \pm \frac{\sqrt{39}i}{2} \).

Step 9: Write the Final Solution

The solutions are \( y = \frac{9}{2} + \frac{\sqrt{39}i}{2} \) and \( y = \frac{9}{2} - \frac{\sqrt{39}i}{2} \).

Key Concepts

Quadratic Equation

A quadratic equation is a mathematical expression of the form \( ax^2 + bx + c = 0 \) where \( a, b, \) and \( c \) are constants, and \( a eq 0 \). This equation represents a parabola on a graph. In our exercise, the equation presented was \( y^2 - 9y + 30 = 0 \). Here, we see:

• \( a = 1 \), indicating that the parabola opens upwards.
• \( b = -9 \), which affects the axis of symmetry and vertex position.
• \( c = 30 \), the y-intercept of the quadratic graph.

For solving quadratic equations, common methods include factoring, using the quadratic formula, or, as in this case, completing the square. Each method has its unique benefits depending on the specific equation. Completing the square is particularly helpful for understanding how the quadratic can be rewritten as a square of a binomial.

Perfect Square Trinomial

A perfect square trinomial is a special form of quadratic expressions. It is written as \((x + d)^2\) or \((x - d)^2\), and when expanded, it equals \(x^2 + 2dx + d^2\). In our scenario, we transformed \(y^2 - 9y\) into a perfect square trinomial.
To do this, we:

• Identified the coefficient of \(y\), which was \(-9\).
• Calculated half of \(-9\), yielding \(-\frac{9}{2}\).
• Squared \(-\frac{9}{2}\) to get \(\frac{81}{4}\).

This value, \(\frac{81}{4}\), was then added to both sides of the equation, turning the expression \(y^2 - 9y + \frac{81}{4}\) into the perfect square \((y - \frac{9}{2})^2\). This process simplifies solving the quadratic equation by making it much easier to see the solutions.

Imaginary Numbers

Imaginary numbers arise when we take the square root of a negative number. In our solved equation, taking the square root of both sides resulted in the expression \( \pm \sqrt{\frac{-39}{4}} \).Since square roots of negative numbers are not defined within the realm of real numbers, we use the imaginary unit \( i \), where \( i = \sqrt{-1} \). This allows us to write the expression as \( \pm \frac{\sqrt{39}i}{2} \).
In conclusion, when solving \((y - \frac{9}{2})^2 = -\frac{39}{4}\), you end up with complex solutions due to this imaginary number. The final solutions are expressed as \( y = \frac{9}{2} \pm \frac{\sqrt{39}i}{2} \).
Understanding imaginary numbers is crucial for solving equations that don't intersect the x-axis, as these imply that our solutions exist as complex numbers with both real and imaginary components.