Problem 13 - Elementary Algebra

Question

Solve the following quadratic equations. \(7 p^{2}+10=26\)

Step-by-Step Solution

Answer

The solutions are \(p = \frac{4 \text{sqrt}(7)}{7}\) and \(p = -\frac{4 \text{sqrt}(7)}{7}\)

Step 1: Simplify the Equation

First, simplify the equation by moving all terms to one side to set it equal to zero. Given equation: \[7 p^{2} + 10 = 26\]Subtract 26 from both sides:\[7 p^{2} + 10 - 26 = 0\]Which simplifies to:\[7 p^{2} - 16 = 0\]

Step 2: Isolate the Quadratic Term

Next, isolate the quadratic term by adding 16 to both sides:\[7 p^{2} = 16\]

Step 3: Solve for the Squared Term

To solve for the squared term, divide both sides by 7:\[p^{2} = \frac{16}{7}\]

Step 4: Take the Square Root

Take the square root of both sides to solve for \(p\):\[p = \frac{\text{sqrt}(16)}{\text{sqrt}(7)}\]This simplifies to:\[p = \frac{4}{\text{sqrt}(7)}\]Rationalize the denominator:\[p = \frac{4 \times \text{sqrt}(7)}{7}\]

Step 5: Consider Both Roots

Remember that taking the square root gives two possible solutions:\[p = \frac{4 \text{sqrt}(7)}{7}\] and \[p = -\frac{4 \text{sqrt}(7)}{7}\]

Key Concepts

Solving Quadratic Equations

To solve a quadratic equation, you follow a series of steps to find the values of the variable that make the equation true. Quadratic equations are in the form \[ax^2 + bx + c = 0\] where \(a\), \(b\), \(c\) are constants. In the example provided, we start by moving all terms to one side of the equation:

• Given equation: \[7p^2 + 10 = 26\]
• Subtract 26 from both sides:\[7p^2 + 10 - 26 = 0\]
• This simplifies to:\[7p^2 - 16 = 0\]

Setting the equation to zero is crucial as it lets us apply various methods for solving, such as factoring, completing the square, or using the quadratic formula.

Simplifying Equations

Simplifying equations is an important step in solving quadratic equations. It often involves combining like terms and simplifying expressions to make them easier to solve. For example:

• Simplified equation: \[7p^2 - 16 = 0\]
• Next step: Add 16 to both sides to isolate the quadratic term:\[7p^2 = 16\]
• Divide by 7 to simplify further:\[p^2 = \frac{16}{7}\]

Square Roots

Taking the square root is a common method to solve for a variable in quadratic equations. Once you isolate the squared term, you can solve for the variable by taking the square root of both sides:

• From the equation: \[p^2 = \frac{16}{7}\]
• Taking the square root of both sides:\[p = \frac{\text{sqrt}(16)}{\text{sqrt}(7)}\]
• This simplifies to:\[p = \frac{4}{\text{sqrt}(7)}\]

Remember to consider both positive and negative roots because the square of both a positive and negative number results in the same value:

Rationalizing the Denominator

Rationalizing the denominator means to eliminate any square roots or irrational numbers from the denominator of a fraction. Here’s how to do it with the given solution:

• We have: \[p = \frac{4}{\text{sqrt}(7)}\]
• To rationalize, multiply the numerator and the denominator by \(\text{sqrt}(7)\):
• This gives us:\[p = \frac{4 \times \text{sqrt}(7)}{7}\]

Now the denominator is rational, and we have the simplified solution:\(p = \frac{4 \text{sqrt}(7)}{7}\), or its negative counterpart:\(p = -\frac{4 \text{sqrt}(7)}{7}\).