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 $1.5$.

This quiz explores the transformation of equations into standard quadratic form and challenges your understanding of solving quadratic equations. You will also test your ability to identify valid steps in solving and interpreting the results of substitutions in equations.