Problem 35 - Elementary Algebra

Question

For the following problems, solve the equations using the quadratic formula. $$ y^{2}=-5 y+4 $$

Step-by-Step Solution

Answer

Answer: The values of y that satisfy the equation are $$y_1 = \frac{-5 + \sqrt{41}}{2}$$ and $$y_2 = \frac{-5 - \sqrt{41}}{2}$$.

Step 1: Rewrite the given equation in standard form

We are given the equation $$y^2 = -5y + 4$$. We can rewrite this equation in the standard form ($$ay^2 + by + c=0$$) by subtracting $$-5y$$ and 4 from both sides. This gives us: $$ y^2 + 5y - 4 = 0 $$

Step 2: Identify the coefficients a, b, and c

Now we can identify the coefficients a, b, and c. In this equation, we have: $$ a=1\\ b=5\\ c=-4 $$

Step 3: Calculate the discriminant

The discriminant is the value inside the square root in the quadratic formula and is calculated as follows: $$ \Delta = b^2 - 4ac $$ Substitute the values of a, b, and c into the formula: $$ \Delta= (5)^2 - 4(1)(-4)=25+16=41 $$

Step 4: Apply the quadratic formula

Now, we'll apply the quadratic formula to find the values of y: $$ y = \frac{-b \pm \sqrt{\Delta}}{2a} $$ Substitute the values of a, b, and Δ: $$ y = \frac{-5 \pm \sqrt{41}}{2(1)} $$ So, the two possible values of y that satisfy the given equation are: $$ y_1 = \frac{-5 + \sqrt{41}}{2} $$ and $$ y_2 = \frac{-5 - \sqrt{41}}{2} $$

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation. It is characterized by having the highest degree term as a square; thus, it takes the general form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). This form is called the standard form.Quadratic equations are quite significant in mathematics as they appear in various fields such as physics, engineering, and economics. They can model various real-world problems, from calculating areas to understanding the motion of objects under the influence of gravity.To solve a quadratic equation, one could use several methods such as:

• Factoring
• Completing the square
• Using the quadratic formula

The quadratic formula offer a universal approach, providing solutions even when other methods are too cumbersome or impractical.

Discriminant

The discriminant is your key to understanding the nature of the roots of a quadratic equation. It's calculated using the formula \(\Delta = b^2 - 4ac\). The value of the discriminant can tell you a lot about the solutions to the quadratic equation.Here's how to interpret the discriminant value:

• If \(\Delta > 0\), the equation has two distinct real roots.
• If \(\Delta = 0\), there is one real repeated root, meaning both solutions are the same.
• If \(\Delta < 0\), the roots are complex numbers, involving imaginary numbers.

Thus, the discriminant not only assists in verifying whether roots are real or complex but also helps in estimating the number of possibly unique solutions.

Polynomial Equations

Polynomial equations, including quadratic equations, are expressions involving sums of powers of variables. The general form of a polynomial is \(ax^n + bx^{n-1} + \ldots + k = 0\), where the degree of the polynomial is \(n\), the highest power of the variable.Polynomials can be classified based on the degree:

• Linear if the degree is 1
• Quadratic if the degree is 2
• Cubic if the degree is 3
• Quartic if the degree is 4

Quadratics are common due to their manageable complexity and because they model many natural phenomena intuitively.In practice, solving polynomial equations involves techniques like factoring, graphing, and using algorithms or formulas tailored to the specific form, such as the quadratic formula for quadratics. Effective problem-solving often requires recognizing patterns and applying the most efficient method for the equation's degree and properties.