Problem 3 - Beginning and Intermediate Algebra

Question

A student solved \(5 x^{2}-5 x+1=0\) incorrectly as follows. $$ \begin{array}{l} x=\frac{-(-5) \pm \sqrt{(-5)^{2}-4(5)(1)}}{2(5)} \\ x=\frac{5 \pm \sqrt{5}}{10} \\ x=\frac{1}{2} \pm \sqrt{5} \end{array} $$

Step-by-Step Solution

Answer

The correct roots are \(x = \frac{1}{2} \pm \frac{\sqrt{5}}{10}\).

Step 1: Identify the Quadratic Formula

The quadratic formula to solve a quadratic equation of the form \(ax^2 + bx + c = 0\) is given by \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].

Step 2: Extract Coefficients

For the equation \(5x^2 - 5x + 1 = 0\), identify the coefficients: \(a = 5\), \(b = -5\), and \(c = 1\).

Step 3: Substitute into the Quadratic Formula

Substitute the coefficients into the quadratic formula: \[x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(5)(1)}}{2(5)}\].

Step 4: Simplify the Expression

Simplify the expression: \[x = \frac{5 \pm \sqrt{25 - 20}}{10}\].

Step 5: Calculate the Discriminant

Calculate the discriminant: \(25 - 20 = 5\). So, \[x = \frac{5 \pm \sqrt{5}}{10}\].

Step 6: Simplify Further

Divide each term by 10: \[x = \frac{5}{10} \pm \frac{\sqrt{5}}{10} \]. Simplify \(\frac{5}{10}\) to \(\frac{1}{2}\), giving \[x = \frac{1}{2} \pm \frac{\sqrt{5}}{10}\].

Step 7: Identify the Error

The student's mistake was in the last simplification step. Correct simplification should give \[x = \frac{1}{2} \pm \frac{\sqrt{5}}{10}\], not \(x = \frac{1}{2} \pm \sqrt{5}\).

Key Concepts

quadratic formula

The quadratic formula is a fundamental tool used to solve quadratic equations of the form \( ax^2 + bx + c = 0 \). This formula connects coefficients from the equation with the solution using the formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].

Here's how it works:

• \