Question

Solve the quadratic equation by finding square roots or by using the quadratic formula. Explain why you chose the method. $$y^{2}-3 y=1$$

Short answer

The solutions to the equation are \(y = \frac{3 + \sqrt{13}}{2}\), and \(y = \frac{3 - \sqrt{13}}{2}\).

Step-by-step solution

Preparation

In general form, a quadratic equation is represented as \(ax^2 + bx + c = 0\). From the quadratic equation to solve \(y^2 - 3y - 1 = 0\), here a=1, b=-3, and c=-1.

Applying the Quadratic Formula

The quadratic formula is given as \(y = \frac{-b \pm \sqrt{b^{2} -4ac}}{2a}\). Substituting the values, \(y = \frac{-(-3) \pm \sqrt{(-3)^{2} -4(1)(-1)}}{2(1)}\). Which simplifies to \(y = \frac{3 \pm \sqrt{9 - (-4)}}{2}\) and then to \(y = \frac{3 \pm \sqrt{13}}{2}\).

Final Solution

Therefore, the solutions to the quadratic equation are \(y = \frac{3 + \sqrt{13}}{2}\), and \(y = \frac{3 - \sqrt{13}}{2}\).