Problem 17 - Algebra and Trigonometry

Question

Solve equation by the square root property. $$ 5 x^{2}+1=51 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = \sqrt{10}\) and \(x = -\sqrt{10}\)

Step 1: Rearrange the equation

The first step is to shift all the terms to one side to isolate the \(x^2\) term. Subtract 1 from both sides of the equation to do this:\(5x^2 = 51 - 1\)which simplifies to:\(5x^2 = 50\)

Step 2: Isolate the \(x^2\) term

Next, divide both sides of the equation by 5:\(\frac{5x^2}{5} = \frac{50}{5}\)This simplifies to:\(x^2 = 10\)

Step 3: Apply the square root property

Applying the square root property to solve for x requires taking the square root of both sides of the equation and considering both the positive and negative roots:\(x = \sqrt{10}, x = -\sqrt{10}\)

Key Concepts

Understanding Quadratic Equations

Quadratic equations are a type of polynomial equation where the highest exponent of the variable is 2. In a general form, they appear as \(ax^2 + bx + c = 0\). The key aspect of quadratic equations is that they can have two solutions, which can be real or complex numbers.
For example, in the equation \(5x^2 + 1 = 51\), it's important to arrange it so that all terms involving the variable \(x\) are on one side. This is the first step to make the equation easier to solve.
Quadratic equations often involve isolating the \(x^2\) term, because this helps in applying techniques like the square root property or the quadratic formula, both of which are valuable for finding solutions.

• A standard form makes it easier to use these methods effectively.
• Quadratics can often be solved by factoring, completing the square, or using the quadratic formula.

The Importance of Isolation of Terms

Isolation of terms is a crucial step in solving equations, especially quadratic ones. This involves rearranging the equation so that the term involving the variable \(x^2\) stands alone on one side of the equation. In our example, the given equation \(5x^2 + 1 = 51\) was rearranged by subtracting 1 from both sides to become \(5x^2 = 50\).
The next step is to isolate \(x^2\) completely. This is done by dividing both sides by the coefficient of \(x^2\), which is 5 in this case, resulting in \(x^2 = 10\). This guarantees that the equation is prepared for the next step and makes it simpler to identify the solutions.
This process significantly simplifies the problem:

• It reduces the complexity of the equation.
• Makes it ready for the application of the square root property or other solving methods.

Exploring Positive and Negative Roots

When solving equations using the square root property, it’s essential to consider both the positive and negative roots. This is because squaring a positive or negative number yields the same positive result. In the equation \(x^2 = 10\), taking the square root of both sides gives us \(x = \sqrt{10}\) and \(x = -\sqrt{10}\).
Though these numbers may seem similar, they represent two different solutions. It's critical to acknowledge both roots because:

• Ignoring one may lead to missing a valid solution.
• Quadratic equations, by nature, can have two solutions.

Understanding this concept ensures a thorough exploration of potential solutions in quadratic problems and provides a complete answer to the problem at hand.