Problem 67 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ 4 x^{2}=81 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{9}{2}\) and \(x = -\frac{9}{2}\).

Step 1: Isolate the Quadratic Term

To solve the equation \(4x^2 = 81\), we first want to isolate the term \(x^2\). We do this by dividing both sides of the equation by 4, which gives us: \[x^2 = \frac{81}{4}\]

Step 2: Take the Square Root

Now that \(x^2\) is isolated, take the square root of both sides to solve for \(x\). Remember that taking the square root of a squared variable results in a positive and a negative solution. \[x = \pm \sqrt{\frac{81}{4}} \]

Step 3: Simplify the Square Root

Let's simplify \(\sqrt{\frac{81}{4}}\). We can separate the square root into two parts (the numerator and the denominator): \[x = \pm \frac{\sqrt{81}}{\sqrt{4}}\] Simplify further by taking the square root of 81 and 4: \[x = \pm \frac{9}{2}\]

Step 4: Write the Solutions

The solutions to the equation are \(x = \frac{9}{2}\) and \(x = -\frac{9}{2}\).

Key Concepts

Isolating Terms

When solving a quadratic equation, one of the first steps is to isolate the term with the variable you want to solve for. This means getting the variable by itself on one side of the equation. In the equation \(4x^2 = 81\), the term we want to isolate is \(x^2\). To do this, you need to remove the 4 that is multiplied by \(x^2\).

One way to isolate the term is by dividing both sides of the equation by 4. This operation helps in canceling out the 4 on the left side. So you perform:

• \(4x^2\) divided by 4 becomes \(x^2\).
• 81 divided by 4 becomes \(\frac{81}{4}\).

After these operations, your equation simplifies to \(x^2 = \frac{81}{4}\). By having \(x^2\) on its own, you can more easily proceed to finding the solution for \(x\). Notice how isolating the term sets the stage for further mathematical operations, like taking the square root.

Square Roots

Once the term \(x^2\) is isolated, the next step is to solve for \(x\) by taking the square root of both sides of the equation. This involves understanding what a square root does. A square root is essentially the inverse of squaring a number.

For the equation \(x^2 = \frac{81}{4}\), taking the square root means you find the number that, when squared, equals \(\frac{81}{4}\). Mathematically, this involves:

• \(x = \pm \sqrt{\frac{81}{4}}\).

The \(\pm\) symbol is crucial here. It indicates that there are two possible solutions because both a positive and a negative number, when squared, give the same result. For example, both \(3^2 = 9\) and \((-3)^2 = 9\).

Taking the square root helps in "undoing" the square in \(x^2\), allowing you to solve for \(x\) directly.

Simplifying Radicals

After taking the square root, you often encounter a radical, which can usually be simplified into a simpler form. Simplifying entails breaking down the radical into more manageable parts. For the expression \(\sqrt{\frac{81}{4}}\), this can be simplified by treating the numerator and denominator separately.

You can write it as:

• \(x = \pm \frac{\sqrt{81}}{\sqrt{4}}\).

Next, you compute the square roots directly:

• \(\sqrt{81} = 9\), since \(9^2 = 81\).
• \(\sqrt{4} = 2\), since \(2^2 = 4\).

Thus, the expression simplifies to \(x = \pm \frac{9}{2}\). By simplifying the radical expression, you convert a complex fraction under a radical into a simpler arithmetic result. This final solution makes interpretation and further calculations much more straightforward.