Problem 83 - Elementary Algebra

Question

Solve using the quadratic formula. $$ -3(y+3)(y-5)=5 y+46 $$

Step-by-Step Solution

Answer

No real solutions; solutions are complex: \(y = \frac{-1 \pm i\sqrt{11}}{-6}\).

Step 1: Expand and Simplify

Start by expanding the left side of the equation: \(-3(y+3)(y-5) = -3(y^2 - 5y + 3y - 15) = -3(y^2 - 2y - 15) = -3y^2 + 6y + 45\).This simplifies to: \(-3y^2 + 6y + 45 = 5y + 46\).

Step 2: Move Terms to One Side

Rearrange the equation by moving all terms to one side to form a quadratic equation:\(-3y^2 + 6y + 45 - 5y - 46 = 0\).Combine like terms to simplify:\(-3y^2 + y - 1 = 0\).

Step 3: Identify Coefficients

Identify the coefficients of the quadratic equation \(ax^2 + bx + c = 0\):\(a = -3\), \(b = 1\), \(c = -1\).

Step 4: Use the Quadratic Formula

The quadratic formula is \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).Substitute the coefficients into the formula:\(y = \frac{-1 \pm \sqrt{1^2 - 4(-3)(-1)}}{2(-3)}\).

Step 5: Simplify Inside the Square Root

First, calculate inside the square root:\(b^2 - 4ac = 1 - 12 = -11\).Thus, the square root becomes \(\sqrt{-11}\), which doesn't yield real solutions.

Step 6: Conclude with Complex Solutions

Since the discriminant \(b^2 - 4ac\) is negative, there are no real solutions.The solutions are complex:\(y = \frac{-1 \pm i\sqrt{11}}{-6}\).

Key Concepts

Understanding the Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations. It applies to any equation in the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(x\) is the variable.
The formula itself is expressed as:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This equation provides solutions to any quadratic equation, whether the solutions are real or complex. The symbol \(\pm\) indicates that there will typically be two solutions: one using a plus sign and one using a minus sign.
By substituting the values of \(a\), \(b\), and \(c\) from your equation into the formula, you can easily find the roots without needing to factor the equation, making it especially helpful when dealing with more complicated expressions.

Exploring Complex Numbers

Complex numbers arise when you have to deal with the square root of negative numbers, something that happens when the discriminant of a quadratic equation is negative.
A complex number is composed of two parts:

• The real part
• The imaginary part

The imaginary part includes the unit \(i\), which is defined as \(i = \sqrt{-1}\). Therefore, if you encounter \(\sqrt{-11}\) in your calculations, this can be expressed as \(i\sqrt{11}\).
In the calculation of quadratic equations, complex solutions occur when there are no real solutions, meaning the graph of the function does not intersect the x-axis.
Complex numbers are fundamental in advanced mathematics and engineering, providing insight into phenomena not evident through real numbers alone.

The Role of the Discriminant

The discriminant is a crucial component of the quadratic formula.

• It is given by the expression \(b^2 - 4ac\).

The value of the discriminant informs us about the nature of the roots of the quadratic equation:

• If \(b^2 - 4ac > 0\), the equation has two distinct real roots.
• If \(b^2 - 4ac = 0\), the equation has exactly one real root, a repeated or "double" root.
• If \(b^2 - 4ac < 0\), the equation has two complex roots.

This information is valuable since it tells us immediately about the type of solutions we can expect, guiding our approach in solving the equation effectively.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\).
Several methods can be employed to solve such equations:

• Factoring, though it is sometimes not feasible if the equation is too complex.
• Completing the square, which can be useful for understanding the vertex form of a quadratic.
• The quadratic formula, which is a universal method suitable for any quadratic equation, especially non-factorable ones.

The quadratic formula, in particular, allows us to find the roots directly by plugging the values of \(a\), \(b\), and \(c\) into the formula. This method ensures accuracy and is indispensable when dealing with complex roots. Whether you're solving for real or complex solutions, understanding these methods gives you a comprehensive toolset to tackle quadratic equations of any form.