Problem 72 - Algebra A Combined Function

Question

Solve each linear or quadratic equation \(4 x^{2}-9=0\)

Step-by-Step Solution

Answer

The solutions are \(x = \frac{3}{2}\) and \(x = -\frac{3}{2}\).

Step 1: Add 9 to both sides of the equation

Start by isolating the quadratic term by adding 9 to both sides of the equation. This gives us:\[4x^2 = 9\]

Step 2: Divide both sides by 4

To solve for \(x^2\), divide both sides of the equation by 4:\[x^2 = \frac{9}{4}\]

Step 3: Take the square root of both sides

To find the value of \(x\), take the square root of both sides of the equation. Remember that taking the square root will introduce both positive and negative solutions:\[x = \pm \sqrt{\frac{9}{4}}\]

Step 4: Simplify the square root

Simplify the expression under the square root:\[x = \pm \frac{3}{2}\]This is because \(\sqrt{9} = 3\) and \(\sqrt{4} = 2\).

Key Concepts

Quadratic Formula

The quadratic formula is a superstar tool in mathematics for solving quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). This method is especially useful when equations are not easily factorable. The formula is:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This looks complex, but let's break it down.
First, \(a\), \(b\), and \(c\) are coefficients from the quadratic equation. The term \(b^2 - 4ac\) is called the discriminant. It determines the number and nature (real or complex) of solutions we get.

The formula gives us solutions directly by substituting the values for \(a\), \(b\), and \(c\). It works reliably for any quadratic equation, making it a must-know technique in algebra.

Square Root Method

The square root method simplifies finding solutions for quadratic equations that don't have an \(x\) term (linear term). For example, with an equation like \(ax^2 + c = 0\), this method is very effective.

• Isolate the \(x^2\) term on one side first to keep things orderly, as seen in the solution \(4x^2 = 9\).
• Then, divide both sides by the coefficient of \(x^2\).
• Finally, take the square root of both sides.
  This will give you both a positive and negative solution, shown here as \(x = \pm \sqrt{\frac{9}{4}}\).The reason we have two solutions is that both the positive and negative numbers squared will give the original number in the equation.

Using the square root method is great when the equation is in the perfect form to take direct square roots.

Isolation of Variable

Isolating the variable is a fundamental step in solving any equation, not just quadratic ones. The goal is to "isolate" the variable you're trying to solve for, typically \(x\), on one side of the equation to simplify the problem significantly.

Here's how it works:

• Start by moving terms that don't contain the variable (constants) to the opposite side of the equation. In the given exercise, you add 9 to both sides to shift the constant term.
• Adjust the equation to have \(x\) terms on one side and everything else on the other. Often, this involves steps like dividing by the coefficient of \(x\). For instance, divide the whole equation by 4.
• Once the variable is isolated, perform operations (like square roots) to solve for the variable completely.
  By isolating the variable, solving the equation becomes a structured and logical process.