Problem 82 - Algebra 2

Question

Solve each equation by using the method of your choice. Find exact solutions. \(2 p^{2}=5 p+6\)

Step-by-Step Solution

Answer

The exact solutions are \(p = \frac{5 + \sqrt{73}}{4}\) and \(p = \frac{5 - \sqrt{73}}{4}\).

Step 1: Rearrange the Equation

First, let's rearrange the equation to set all terms on one side and equal to zero. This will give us a quadratic equation that has the form: \[ 2p^2 - 5p - 6 = 0 \]

Step 2: Identify Coefficients

Identify the coefficients from the quadratic equation, which has the standard form \(ax^2 + bx + c = 0\). Here, \(a = 2\), \(b = -5\), and \(c = -6\).

Step 3: Apply the Quadratic Formula

Use the quadratic formula to find the roots of the equation: \[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Substituting the coefficients into the formula gives:\[ p = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 2 \cdot (-6)}}{2 \cdot 2} \]

Step 4: Simplify Under the Square Root

Calculate the expression under the square root (the discriminant):\[ (-5)^2 - 4 \cdot 2 \cdot (-6) = 25 + 48 = 73 \]

Step 5: Calculate the Solutions

Plug the simplified discriminant back into the quadratic formula:\[ p = \frac{5 \pm \sqrt{73}}{4} \]This gives us the two solutions:\[ p_1 = \frac{5 + \sqrt{73}}{4} \] \[ p_2 = \frac{5 - \sqrt{73}}{4} \]

Step 6: Write the Exact Solutions

The exact solutions to the equation \(2p^2 = 5p + 6\) are:\[ p = \frac{5 + \sqrt{73}}{4} \] and \[ p = \frac{5 - \sqrt{73}}{4} \]

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). It's especially helpful when factoring is difficult or impossible. The formula is given by:\[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula can find the roots (solutions) of any quadratic equation by substituting the coefficients \(a\), \(b\), and \(c\) into it.

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

The '+/-' symbol means that there will be two solutions: one using the plus sign and one using the minus sign.
To use the quadratic formula correctly, it's important to first ensure your equation is set to zero, which turns it into a standard quadratic form. After that, identifying the correct coefficients is essential.

Discriminant

The discriminant is a key part of the quadratic formula, located under the square root symbol. It is calculated as:\[ b^2 - 4ac \]The discriminant helps us determine the number and type of solutions for a quadratic equation.

• If the discriminant is positive, there are two distinct real roots.
• If the discriminant is zero, there is exactly one real root (or a repeated root).
• If the discriminant is negative, there are no real roots, but two complex roots.

In this problem, the discriminant is 73, which is positive, indicating that there are two distinct real solutions. Notice that you calculate it before proceeding with the rest of the quadratic formula. This early insight can save time and inform you of the nature of the solutions you will find.

Solving Equations

Solving equations involves finding the value(s) of the variable that satisfy the equation. When solving quadratic equations, one common method is to rearrange the terms so that they equal zero, thus forming a standard quadratic equation.
After the rearrangement, identifying the coefficients helps in using methods like factoring, completing the square, or, as discussed here, applying the quadratic formula.
By substituting these coefficients into the quadratic formula, we solve for the variable \(p\) in this case. The step-by-step approach ensures each calculation is carried out meticulously, minimizing errors and leading to the correct solutions.

Roots of Equations

Roots of a quadratic equation are the solutions that satisfy the equation. In simpler terms, they are the values of \(p\) that make the equation zero. The two roots can be found by calculating them from the quadratic formula results:\[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]For each equation, there can be two, one, or no real roots, as revealed by the discriminant. In this particular problem:

• The roots are \( \frac{5 + \sqrt{73}}{4} \) and \( \frac{5 - \sqrt{73}}{4} \).
• The positive discriminant means these are two distinct real roots.

Finding these roots is crucial in understanding the behavior of the quadratic function, especially in graphing or applications involving maximum or minimum values.