Problem 88 - Elementary Algebra

Question

Solve using the quadratic formula. $$ 3(t-3)-t(t-5)=7 t $$

Step-by-Step Solution

Answer

The solutions are complex: \( t = \frac{1 + i\sqrt{35}}{2} \) and \( t = \frac{1 - i\sqrt{35}}{2} \).

Step 1: Expand and Simplify the Equation

First, expand the expression and simplify the equation. Start by distributing and reorganizing:\[ 3(t - 3) - t(t - 5) = 7t \]\[ 3t - 9 - t^2 + 5t = 7t \]Combine like terms:\[ -t^2 + 8t - 9 = 7t \]

Step 2: Move All Terms to One Side

Move all the terms to one side of the equation to form a quadratic equation.\[ -t^2 + 8t - 9 - 7t = 0 \]Combine the terms to simplify further:\[ -t^2 + t - 9 = 0 \]

Step 3: Rearrange and Set in Standard Form

Rearrange the equation into the standard form of a quadratic equation, \( at^2 + bt + c = 0 \).\[ -t^2 + t - 9 = 0 \] becomes:\[ t^2 - t + 9 = 0 \] by multiplying through by -1.

Step 4: Identify Coefficients

Identify the coefficients \( a \), \( b \), and \( c \) from the quadratic equation:\( a = 1 \), \( b = -1 \), \( c = 9 \).

Step 5: Apply the Quadratic Formula

Apply the quadratic formula:\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Substitute \( a = 1 \), \( b = -1 \), \( c = 9 \):\[ t = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot 9}}{2 \cdot 1} \]\[ t = \frac{1 \pm \sqrt{1 - 36}}{2} \]

Step 6: Simplify the Solution

Simplify the term inside the square root:\[ t = \frac{1 \pm \sqrt{-35}}{2} \]Since the discriminant \(-35\) is negative, the roots are complex.

Step 7: Write the Solution in Complex Form

Express the solution using imaginary numbers:\[ t = \frac{1 \pm i\sqrt{35}}{2} \]This represents two complex solutions.

Key Concepts

Complex Solutions

When dealing with quadratic equations, the concept of complex solutions arises when the discriminant, the part under the square root in the quadratic formula, is negative. The discriminant is defined as \( b^2 - 4ac \).
In the context of our exercise, the discriminant was calculated as \( 1 - 36 = -35 \). This negative discriminant indicates that the solutions are not real numbers but rather complex numbers. Complex numbers are expressed in the form \( a + bi \), where \( i \) is the imaginary unit, equal to the square root of -1.

• When the discriminant is negative, the quadratic equation has two conjugate complex solutions.
• The solutions will be of the form \( \frac{-b \, \pm \, i\sqrt{|b^2 - 4ac|}}{2a} \).

In this problem, the solutions are \( \frac{1 \pm i\sqrt{35}}{2} \). This is because the imaginary unit \( i \sqrt{35} \) appears due to the square root of the negative discriminant.

Quadratic Equation

The quadratic equation is a key concept in algebra and forms the basis of this particular exercise. It is typically presented in the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \).
For the given problem, initially, we expanded and simplified to obtain \( -t^2 + t - 9 = 0 \). By multiplying through by -1, it was rewritten to \( t^2 - t + 9 = 0 \) to meet the standard form.

• The quadratic equation represents a parabola when plotted on a graph.
• The solutions to the equation are the points at which the parabola intersects the x-axis. For complex solutions, the parabola does not touch the x-axis.

Understanding how to structure a quadratic equation is fundamental in solving it using the quadratic formula.

Discriminant

The discriminant is a pivotal component in solving quadratic equations using the quadratic formula. It helps determine the nature and number of solutions. The discriminant is given by \( b^2 - 4ac \).
In this exercise, the coefficients \( a = 1 \), \( b = -1 \), and \( c = 9 \) were used in calculating the discriminant as \( (-1)^2 - 4 \times 1 \times 9 = 1 - 36 = -35 \). This result determines that the equation has complex solutions.

• If the discriminant is positive, the quadratic equation has two distinct real solutions.
• If it is zero, the equation has exactly one real solution (or a repeated real solution).
• When negative, as in this problem, it leads to two complex conjugate solutions.

Thus, the discriminant gives crucial information about the nature of the quadratic equation's solutions, guiding the process of solving it.