Problem 135 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The solution to the equation \(x(x-7)=3\) are x = \( (7 + \sqrt{61}) / 2\), x = \( (7 - \sqrt{61})/2\)

Step 1: Rewrite the Equation in Standard Quadratic Form

Rearrange the equation x(x - 7) = 3 to look like a standard quadratic equation. This can be done by expanding the brackets to get \(x^2 - 7x = 3\). Then, for ax^2 + bx + c = 0, rearrange the equation to get \(x^2 - 7x - 3 = 0\).

Step 2: Identify the values of a, b, and c

In this equation \(x^2 - 7x - 3 = 0\), a is the coefficient of \(x^2\), b is the coefficient of x, and c is the constant term. Therefore, a = 1, b = -7, and c = -3.

Step 3: Apply the Quadratic Formula

The quadratic formula is \(x = [ -b \pm \sqrt{b^2 - 4ac} ] / (2a)\). With the values of a, b, and c we have, plug these into the formula to find the solution.

Step 4: calculate the equation for x

Substituting the values in gives \(x = ( 7 \pm \sqrt{((-7)^2 - 4 * 1 * -3) }) / (2 * 1)\) = \( (7 \pm \sqrt{49-(-12)}) / 2 = (7 \pm \sqrt{61})/2\)

Step 5: Finalize the solution

The equation \(x = (7 \pm \sqrt{61})/2\) gives two possible solutions for x, \(x = (7 + \sqrt{61})/2\) and \(x = (7 - \sqrt{61})/2\)