Problem 45 - Algebra A Combined Function

Question

Use the discriminant to determine the number and types of solutions of each equation. $$ 3 x=-2 x^{2}+7 $$

Step-by-Step Solution

Answer

The equation has two distinct irrational solutions.

Step 1: Write the Equation in Standard Form

First, we rewrite the given equation \(3x = -2x^2 + 7\) in the standard quadratic form \(ax^2 + bx + c = 0\). To do this, move all terms to one side of the equation: \(2x^2 + 3x - 7 = 0\).

Step 2: Identify Coefficients

Now that we have the quadratic equation in standard form \(2x^2 + 3x - 7 = 0\), identify the coefficients: \(a = 2\), \(b = 3\), and \(c = -7\).

Step 3: Calculate the Discriminant

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \(D = b^2 - 4ac\). Substitute the values of \(a\), \(b\), and \(c\) into this formula: \(D = 3^2 - 4 \times 2 \times (-7)\).

Step 4: Simplify the Discriminant

Calculate the discriminant: \(D = 9 + 56 = 65\).

Step 5: Determine the Number and Types of Solutions

Since the discriminant \(D = 65\) is positive and not a perfect square, the quadratic equation \(2x^2 + 3x - 7 = 0\) has two distinct irrational solutions.

Key Concepts

Quadratic Equation

A quadratic equation is a specific type of polynomial equation. It takes the form \[ax^2 + bx + c = 0\]where:

• \(a\), \(b\), and \(c\) are constants with \(a eq 0\)
• \(x\) represents an unknown or variable

The hallmark of a quadratic equation is its highest degree, which is two (hence the term "quadratic").
This means the variable \(x\) is always raised to the power of two at most. Quadratic equations represent a parabola when graphed.
They arise in various scenarios, from physics problems involving projectile motion to financial calculations of profit maximization.

Standard Form

The standard form of a quadratic equation is crucial for analysis and solving. This form arranges all terms to be equal to zero:\[ax^2 + bx + c = 0\]Having this specific arrangement allows for easier application of mathematical formulas, such as the quadratic formula or the discriminant.
To convert any given quadratic expression into the standard form, you need to ensure all terms are on one side of the equation.
In the example problem, altering the equation from \(3x = -2x^2 + 7\) to \(2x^2 + 3x - 7 = 0\) is essential. This conversion is the first step in pinpointing important components such as coefficients.

Coefficients

In a quadratic equation, coefficients are the numerical factors linked to the variables, denoted by \(a\), \(b\), and \(c\) in the standard form \[ax^2 + bx + c = 0\]

• \(a\) is the coefficient of the quadratic term \(x^2\)
• \(b\) is the coefficient of the linear term \(x\)
• \(c\) is the constant term

Understanding these coefficients is vital because they directly influence the shape and position of the parabola on a graph.
In the provided example, the coefficients are \(a = 2\), \(b = 3\), and \(c = -7\).
Identifying them helps in calculating the discriminant later, which leads to determining the nature and number of solutions.

Types of Solutions

The types of solutions for a quadratic equation depend on the value of the discriminant. The discriminant, \(D\), is calculated using:\[D = b^2 - 4ac\]The nature of the solutions can be determined as follows:

• If \(D > 0\), the equation has two distinct real solutions. If \(D\) is also a perfect square, these solutions are rational.
• If \(D = 0\), there is exactly one real solution, called a double root. This means the parabola touches the \(x\)-axis at one point.
• If \(D < 0\), the equation has two complex conjugate solutions. The parabola does not intersect the \(x\)-axis at all.

In our example, the discriminant is \(65\), a positive number and not a perfect square.
Thus, the quadratic equation \(2x^2 + 3x - 7 = 0\) yields two distinct irrational solutions.
Understanding these distinctions helps in predicting the behavior of the graph and the roots of the equation.