Problem 138 - Algebra and Trigonometry

Question

Will help you prepare for the material covered in the next section. Solve: \(x(x-7)=3\)

Step-by-Step Solution

Answer

The solutions to the equation \(x(x-7) = 3\) are \(x = 3.5 + \sqrt{15.25}\) and \(x = 3.5 - \sqrt{15.25}\).

Step 1: Rearrange the equation to the form \(ax^2 + bx + c = 0\)

Rearrange the original equation \(x(x-7) = 3\) to the standard form of quadratic equation. This can be done by expanding the brackets and by bringing all terms to the same side. So, it becomes: \(x^2 - 7x - 3 = 0\).

Step 2: Identify the coefficients \(a\), \(b\), and \(c\)

In our quadratic equation \(x^2 - 7x - 3 = 0\), the coefficients are \(a=1\) (the coefficient of \(x^2\)), \(b=-7\) (the coefficient of \(x\)) and \(c=-3\), the constant term.

Step 3: Apply the Quadratic Formula

The quadratic formula is given by \(x = [-b ± \sqrt{b^2 - 4ac}] / (2a)\). Substituting \(a=1\), \(b=-7\) and \(c=-3\) we get: \(x = [7 ± \sqrt{(-7)^2 - 4*1*(-3)}] / (2*1) = [7 ± \sqrt{49 + 12}] / 2 = [7 ± \sqrt{61}] / 2 \).

Step 4: Simplify the Expression

Simplifying the above expression gives us the two roots of the equation which are \(x = 3.5 + \sqrt{15.25}\) and \(x = 3.5 - \sqrt{15.25}\).