Algebra Class: Quadratic Equations

Transforming the Equation

• The equation can be transformed into a standard quadratic form by multiplying both sides by $x$ and subtracting $1.5x$ from both sides. This yields the equation: $x^2 - 1.5x - 1 = 0$
• To solve for x, the quadratic formula can be used, which states that for a quadratic equation in the form $ax^2 + bx + c = 0$, the solutions for x are given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Solving for x

• Plugging the coefficients from the transformed equation into the quadratic formula yields $x = \frac{1.5 \pm \sqrt{1.5^2 + 4}}{2}$.
• The positive root of this equation is the solution, therefore, $x = \frac{1.5 + \sqrt{1.5^2 + 4}}{2}$.
• The solution, $x = \frac{1.5 + \sqrt{1.5^2 + 4}}{2}$ satisfies the equation $x^2 - \frac{1}{x} = 1.5$.

Invalid Step

• An invalid step when solving $\frac{x^2 - 1}{x} = 1.5$ is directly multiplying both sides of the equation by $x$ without considering the restriction that $x \neq 0$.
• Multiplying by $x$ leads to the quadratic equation $x^2 - 1.5x - 1 = 0$, which is valid but it introduced a potential solution of $x = 0$ that is not allowed in the original equation.

Substitution

• Substituting $x = -1$ into the equation $\frac{x^2 - 1}{x} = 1.5$ gives us: $\frac{(-1)^2 - 1}{-1} = \frac{1-1}{-1} = 0$.
• Therefore, when $x = -1$ is substituted, the result is $0$.

Right Side of the Equation

• When $\frac{x^2 - 1}{x} = 1.5$ is simplified, the right side of the equation becomes $1.5$.