Problem 47 - Elementary Algebra

Question

Choose the appropriate method to solve the following. $$ 9 t 2=4(3 t-1) $$

Step-by-Step Solution

Answer

The solution is t = 2/3.

Step 1: Expand the Equation

Start by distributing the number on the right side of the equation. Distribute 4 to both terms inside the parentheses. This gives:\[4(3t - 1) = 4 \times 3t - 4 \times 1 = 12t - 4\]

Step 2: Rewrite the Equation

Now, substitute the expanded terms back into the original equation:\[9t^2 = 12t - 4\]This equation is now a quadratic equation.

Step 3: Set the Equation to Zero

Rearrange all terms to one side of the equation so it equals zero:\[9t^2 - 12t + 4 = 0\]

Step 4: Use the Quadratic Formula

The quadratic formula is:\[t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]In this equation, \(a = 9\), \(b = -12\), and \(c = 4\). Substitute these values into the formula.

Step 5: Calculate the Discriminant

The discriminant is the part under the square root in the quadratic formula: \(b^2 - 4ac\).\[(-12)^2 - 4 \times 9 \times 4 = 144 - 144 = 0\]The discriminant is 0, indicating a double root.

Step 6: Solve for t Using the Quadratic Formula

Since the discriminant is zero, the equation has one solution:\[t = \frac{-(-12) \pm \sqrt{0}}{2 \times 9} = \frac{12}{18} = \frac{2}{3}\]

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are equations in the form of \(ax^2 + bx + c = 0\). These equations contain a variable raised to the second power. The quadratic formula is:\[t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula provides solutions for any quadratic equation.
Use this formula when:

• The equation cannot be factored easily.
• The equation involves complicated numbers.

It allows you to find both potential solutions for the equation by determining the roots. By plugging in values for \(a\), \(b\), and \(c\) from the equation into the formula, you can directly calculate the values of the variable, in this case, \(t\). This formula will always deliver reliable results for quadratic equations and is especially useful when other methods fail.

Discriminant

The discriminant is an integral part of the quadratic formula and plays a key role in determining the nature of the roots of a quadratic equation. It is denoted by \(b^2 - 4ac\). This value helps us understand what type of solutions to expect.

• If the discriminant is positive (greater than zero), you'll have two distinct real roots.
• If it equals zero, there is exactly one real root, typically called a double root or a repeated root.
• If the discriminant is negative, the equation has no real solutions but rather two complex numbers as roots.

In our example equation, the discriminant was calculated as zero. This informed us that the solution would be a single, repeated real root, which simplifies the process of solving the equation.

Solving Equations

Solving quadratic equations involves a few fundamental steps to isolate the variable, which results in finding one or more solutions.
Here's a general strategy:

• First, ensure the equation is set to zero. This standard form is crucial for using the quadratic formula conveniently.
• Rearrange or simplify the equation if necessary, by expanding expressions or combining like terms, as seen in the given exercise.
• Once simplified, apply the quadratic formula and identify \(a\), \(b\), and \(c\) for substituting into the formula.
• Calculate the discriminant to assess the nature and number of solutions.
• Follow through the arithmetic carefully to reach the solutions after substitution.

This structured methodical approach ensures you can tackle various forms of quadratic equations efficiently. Armed with these basic steps, you can solve equations more confidently and accurately.