Problem 48 - Elementary Algebra

Question

Use the quadratic formula to solve the following. $$-3 y_{2}+7 y=0$$

Step-by-Step Solution

Answer

The solutions are \( y = 0 \) and \( y = \frac{7}{3} \).

Step 1: Identify the coefficients

First, rewrite the quadratic equation in standard form if necessary. The standard form of a quadratic equation is \(ay^2 + by + c = 0\). For the given equation, \(-3y^2 + 7y = 0\), we can identify the coefficients as follows: \(a = -3\), \(b = 7\), and \(c = 0\).

Step 2: Recall the quadratic formula

The quadratic formula is used to find the roots of a quadratic equation, and it is given by: \[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Step 3: Substitute the coefficients into the formula

Substitute \(a = -3\), \(b = 7\), and \(c = 0\) into the quadratic formula: \[ y = \frac{-7 \pm \sqrt{7^2 - 4(-3)(0)}}{2(-3)} \]

Step 4: Simplify under the square root

Calculate the discriminant under the square root. \[ 7^2 - 4(-3)(0) = 49 \] So the equation becomes: \[ y = \frac{-7 \pm \sqrt{49}}{-6} \]

Step 5: Evaluate the square root

Calculate the square root of 49: \[ \sqrt{49} = 7 \] Update the equation to: \[ y = \frac{-7 \pm 7}{-6} \]

Step 6: Solve for the two potential solutions

Separate into two equations to solve:1. \( y = \frac{-7 + 7}{-6} = \frac{0}{-6} = 0 \)2. \( y = \frac{-7 - 7}{-6} = \frac{-14}{-6} = \frac{7}{3} \) Thus, the two solutions are \( y = 0 \) and \( y = \frac{7}{3} \).

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of the second degree, meaning it includes at least one term that is squared. It generally takes the form:

• \( ax^2 + bx + c = 0 \)

where \( a \), \( b \), and \( c \) are constants, and \( x \) represents an unknown variable. Quadratic equations are essential in various mathematical applications, including physics, engineering, and economics, due to their ability to model parabolic relationships.

When the term \( ax^2 \) is present, it assures the equation describes a parabola in a graphical sense, which either opens upwards or downwards depending on the sign of \( a \). Finding the solutions to such equations is a critical mathematical skill and can reveal key characteristics of the mathematical model being studied.

Coefficients

Coefficients in a quadratic equation are the numerical parts of each term. Specifically, in the equation \( ax^2 + bx + c = 0 \):

• \( a \) is the coefficient of the squared term \( x^2 \).
• \( b \) is the coefficient of the linear term \( x \).
• \( c \) is the constant term (or the term without \( x \)).

These coefficients determine the shape and position of the parabola represented by the quadratic equation. For instance, the coefficient \( a \) affects how "wide" or "narrow" the graph is and indicates if it faces upwards (if \( a > 0 \)) or downwards (if \( a < 0 \)).

Understanding the roles these coefficients play is crucial for solving the equation. By substituting these values into specific formulas, like the quadratic formula, one can find the solutions to the equation.

Discriminant

The discriminant is a part of the quadratic formula that gives insight into the nature and number of solutions for a quadratic equation. It is found in the expression:

• \( b^2 - 4ac \)

This expression is located under the square root in the quadratic formula:

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

The value of the discriminant can tell us three things:

• If it is positive, there are two different real solutions.
• If it is zero, there is exactly one real solution.
• If it is negative, there are no real solutions, but two complex solutions.

By calculating the discriminant before finding the square root, we can predict the nature of the solutions without computing them, which is a valuable check during problem-solving.

Solutions of a Quadratic Equation

The solutions of a quadratic equation, also known as its roots, are the values that satisfy the equation (make the equation true). Using the quadratic formula:

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

we can calculate these roots accurately. After substituting the coefficients \( a \), \( b \), and \( c \) into the formula, the key steps involve:

• Calculating the discriminant \( b^2 - 4ac \).
• Finding the square root of the discriminant.
• Determining the two possible values using \( \pm \) in the formula.

These steps yield two potential solutions, which can be either real or complex, depending on the discriminant's value. In our case of \(-3y^2 + 7y = 0\), the solutions are straightforward, yielding \( y = 0 \) and \( y = \frac{7}{3} \), emphasizing the practical utility of the quadratic formula for real-world applications.