Problem 41 - Algebra 2

Question

Solve each equation by using the method of your choice. Find exact solutions. \(3 x^{2}-10 x=7\)

Step-by-Step Solution

Answer

The exact solutions are \(x_1 = \frac{5 + \sqrt{46}}{3}\) and \(x_2 = \frac{5 - \sqrt{46}}{3}\).

Step 1: Rearrange the Equation

Start by moving all terms to one side of the equation to set it equal to zero. The given equation is \(3x^2 - 10x = 7\). To rearrange, subtract 7 from both sides, resulting in the equation \(3x^2 - 10x - 7 = 0\).

Step 2: Identify Coefficients for the Quadratic Formula

To solve the quadratic equation \(3x^2 - 10x - 7 = 0\), identify the coefficients needed for the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 3\), \(b = -10\), and \(c = -7\).

Step 3: Calculate the Discriminant

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(b^2 - 4ac\). Calculate the discriminant: \((-10)^2 - 4(3)(-7) = 100 + 84 = 184\).

Step 4: Solve Using the Quadratic Formula

Substitute the coefficients and the discriminant into the quadratic formula: \[x = \frac{-(-10) \pm \sqrt{184}}{2(3)} = \frac{10 \pm \sqrt{184}}{6}\].The expression \(\sqrt{184}\) simplifies to \(2\sqrt{46}\), thus \[x = \frac{10 \pm 2\sqrt{46}}{6}\], which can be simplified to \[x = \frac{5 \pm \sqrt{46}}{3}\].

Step 5: Determine the Exact Solutions

The exact solutions to the equation \(3x^2 - 10x - 7 = 0\) are \(x_1 = \frac{5 + \sqrt{46}}{3}\) and \(x_2 = \frac{5 - \sqrt{46}}{3}\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find solutions to quadratic equations of the form \(ax^2 + bx + c = 0\). When you have an equation set up like this, you can substitute the values of \(a\), \(b\), and \(c\) into the formula

1. The formula is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
2. This formula will yield two solutions or roots for \(x\), which are denoted by \(x_1\) and \(x_2\).

To use it effectively, you start by identifying the coefficients:

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

In our equation, \(3x^2 - 10x - 7 = 0\), we identified \(a = 3\), \(b = -10\), and \(c = -7\). Substituting these into the quadratic formula lets us compute the solutions for \(x\). Understanding this formula is crucial because it offers a straightforward method for solving any quadratic equation.

Discriminant

The discriminant is a key concept in understanding quadratic equations. It is calculated from the coefficients of the quadratic equation as shown in the formula: \[b^2 - 4ac\]. The discriminant tells you how many and what type of solutions you can expect from the quadratic equation:

• If the discriminant is positive, as it is in this case (184), there are two distinct real solutions.
• If it equals zero, there is one repeated real solution.
• If it is negative, there are two complex solutions, and no real solutions exist.

In the provided example, we calculated the discriminant: \[(-10)^2 - 4(3)(-7) = 100 + 84 = 184\].This positive discriminant confirms that the quadratic equation has two distinct real roots. Knowing this helps you understand the nature of the solutions, even before solving, and provides insight into the graph of the quadratic function.

Exact Solutions

Exact solutions refer to the solutions of the quadratic equation expressed in their most precise form without decimal approximations. When using the quadratic formula, once the discriminant is inserted, solutions can be determined:

• By simplifying the expression inside the square root, if possible. In our case, \(\sqrt{184}\) was simplified to \(2\sqrt{46}\).
• Subsequently simplifying the entire expression: \[x = \frac{10 \pm 2\sqrt{46}}{6}\].

This can then be broken down to: \[x = \frac{5 \pm \sqrt{46}}{3}\]. Exact solutions are often preferable in mathematical studies as they provide precise values, unaffected by rounding errors inherent with decimal forms.These exact forms can give clearer insights into the behavior of functions, allowing for more accurate algebraic manipulation and interpretation. In this specific problem, the solutions obtained were: \(x_1 = \frac{5 + \sqrt{46}}{3}\) and \(x_2 = \frac{5 - \sqrt{46}}{3}\), showcasing the two different intersection points of the quadratic function with the \(x\)-axis.