Problem 48 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ 7 p^{2}-5=11 $$

Step-by-Step Solution

Answer

p = \pm \frac{4 \sqrt{7}}{7}

Step 1 - Isolate the variable term

First, get the term with the variable squared by itself on one side of the equation. Start by adding 5 to both sides of the equation: \[ 7 p^{2} - 5 + 5 = 11 + 5 \]This simplifies to:\[ 7 p^{2} = 16 \]

Step 2 - Divide by the coefficient of the squared term

Next, divide both sides of the equation by 7 to isolate \( p^{2} \):\[ \frac{7 p^{2}}{7} = \frac{16}{7} \]This simplifies to:\[ p^{2} = \frac{16}{7} \]

Step 3 - Apply the square root property

To solve for \( p \), take the square root of both sides of the equation. Remember to include both the positive and negative roots:\[ p = \pm \sqrt{\frac{16}{7}} \]Simplify the radical:\[ p = \pm \frac{\sqrt{16}}{\sqrt{7}} = \pm \frac{4}{\sqrt{7}} \]

Step 4 - Rationalize the denominator

The expression can be simplified further by rationalizing the denominator:\[ p = \pm \frac{4}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} = \pm \frac{4 \sqrt{7}}{7} \]So, the simplified solution is:\[ p = \pm \frac{4 \sqrt{7}}{7} \]

Key Concepts

solving quadratic equations

Quadratic equations are those that include a variable raised to the second power, such as the equation given: $$ 7p^2 - 5 = 11 $$. Solving quadratic equations can often involve isolating the variable term, applying properties of square roots, and simplifying the resulting radicals. The general form of a quadratic equation is:$$ ax^2 + bx + c = 0 $$. In cases where you can express the quadratic equation in the form $$ ax^2 = d $$, you can use the square root property to solve it. The square root property states that for any nonnegative number $$ d $$, if $$ x^2 = d $$, then:$$ x = \pm\sqrt{d} $$. This tells us that there are two potential solutions for $$ x $$ corresponding to the positive and negative square roots of $$ d $$.

simplifying radicals

Simplifying radicals helps to make expressions easier to understand and use. Take the example:$$ p = \pm\frac{\sqrt{16}}{\sqrt{7}} $$Here, $$ \sqrt{16} $$ simplifies neatly to 4 because 16 is a perfect square.This leaves us with:$$ p = \pm\frac{4}{\sqrt{7}} $$. However, $$ \sqrt{7} $$ cannot be simplified into a whole number because 7 is not a perfect square. Therefore, $$ \sqrt{7} $$ remains as is. Understanding which numbers can be simplified and carrying out these simplifications are essential skills in algebra.

rationalizing the denominator

Rationalizing the denominator means rewriting a fraction so that the denominator no longer contains any radicals. Radicals in the denominator are generally undesirable because they can make further arithmetic unwieldy. To rationalize the denominator of:$$ p = \pm\frac{4}{\sqrt{7}} $$, you multiply both the numerator and the denominator by the radical in the denominator, $$ \sqrt{7} $$. This gives:$$ p = \pm\frac{4}{\sqrt{7}}\times\frac{\sqrt{7}}{\sqrt{7}} = \pm\frac{4\sqrt{7}}{7} $$. This process eliminates the radical from the denominator, resulting in a more conventional and simpler format.

isolate variable term

The first step in solving many quadratic equations, including using the square root property, is to isolate the variable term. Starting from our example:$$ 7p^2 - 5 = 11 $$, you first add 5 to both sides to get:$$ 7p^2 = 16 $$. Next, divide both sides by the coefficient of the squared term (7 in this case):$$ p^2 = \frac{16}{7} $$. Through these steps, we isolate the variable term, making it easier to apply the square root property and find the solutions for $$ p $$. Always aim to get the term with the variable on one side and everything else on the other side. This strategy makes solving the equation more straightforward.