Problem 58 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solving an Equation Involving Fractions Find all solutions of the equation. Check your solutions. $$8\left(\frac{t}{t-1}\right)^{2}-2\left(\frac{t}{t-1}\right)-3=0$$

Step-by-Step Solution

Answer

The solution to the equation \(8\left(\frac{t}{t-1}\right)^{2}-2\left(\frac{t}{t-1}\right) - 3 = 0\) is the solution for \(t\) that satisfy both the original equation and are valid for each value of \(a\).

Step 1: Isolate the variable expression

Start by isolating the expression \( \frac{t}{t-1} \). Get all of the terms involving \( \frac{t}{t-1} \) on the left, and the constant on the right side. The equation becomes :\(8\left(\frac{t}{t-1}\right)^{2}-2\left(\frac{t}{t-1}\right) + 3 = 0\)

Step 2: Substitute and apply the quadratic formula

Now, it makes the equation easier to solve if you substitute \( a = \frac{t}{t-1} \). Therefore, our equation is simplified to: \(8a^{2} - 2a + 3 = 0\).Apply the quadratic formula \( a = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\) to solve for \(a\), where \(a = 8\), \(b = -2\), and \(c = 3\). Calculate the discriminant \(D = b^2 - 4ac\), then find the possible values of \(a\).

Step 3: Solve for \(t\)

Now substitute back the expression for \(a = \frac{t}{t-1}\) with each possible value of \(a\) you found in Step 2. Solve for \(t\) from the equation \(t = a(t-1)\) for each possible value of \(a\).

Step 4: Check solutions

Plugging your answer(s) for \(t\) back into the original equation will allow you to check your work. The solutions of \(t\) are the solution of the original equation if and only if the original equation is satisfied.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find solutions to quadratic equations, which are equations in the form of \( ax^2 + bx + c = 0 \). It provides a direct method to calculate the roots of the equation without completing the square or factoring. The formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

• \( a \), \( b \), and \( c \) are coefficients from the quadratic equation.
• The symbol \( \pm \) indicates there could be two possible solutions: one using addition and one using subtraction.

In the exercise, after substituting \( a = \frac{t}{t-1} \), the revised equation \( 8a^{2} - 2a + 3 = 0 \) matches the standard form perfectly. The quadratic formula is then applied to solve for \( a \), showing its flexibility in solving problems involving more complex expressions.

Discriminant

The discriminant, noted as \( D \), is part of the quadratic formula under the square root, specifically \( b^2 - 4ac \). It helps determine how many real roots a quadratic equation has.

• If \( D > 0 \), there are two distinct real roots.
• If \( D = 0 \), there is one real root (a repeated root).
• If \( D < 0 \), there are no real roots (the roots are complex numbers).

In the example given, calculating the discriminant helps identify whether there are real solutions for the substituted equation \( 8a^2 - 2a + 3 = 0 \). Understanding \( D \) offers insight into solution possibilities before further solving steps.

Substitution Method

The substitution method can simplify solving complex equations by replacing more cumbersome expressions with a single variable. This method is handy for dealing with nested fractions and higher-order terms.In the provided solution, substituting \( a = \frac{t}{t-1} \) turns the fractional equation into a more familiar quadratic form. After finding \( a \), substitute back to resolve for \( t \).This approach significantly streamlines solving processes, reducing mistakes and simplifying calculations.

Fractional Equations

Fractional equations involve expressions in the form of fractions and can appear daunting initially. The key is converting them into a more manageable form, similar to polynomial equations.In this exercise, the fraction \( \frac{t}{t-1} \) was effectively handled by turning it into a simpler variable. By doing so, the equation changes from a fractional form to a standard quadratic one, easier to solve using familiar methods like the quadratic formula.Understanding how to tackle fractional equations is crucial for avoiding errors and ensuring correct computations in more complex scenarios.