Question

Solve the quadratic equation by finding square roots or by using the quadratic formula. Explain why you chose the method. $$-2 x^{2}+6 x+1=0$$

Short answer

The solutions to the quadratic equation -2x^2+6x+1=0 by using the quadratic formula are \(x_1=\frac{3 - \sqrt{11}}{2}\) and \(x_2=\frac{3 + \sqrt{11}}{2}\). The quadratic formula was chosen due to the discriminant not being a perfect square.

Step-by-step solution

Identify Coefficients

Identify the coefficients of the quadratic equation -2x^2+6x+1=0. Here, a=-2, b=6, and c=1.

Calculate Discriminant

Calculate the discriminant using the formula b^2 - 4ac. Substituting the coefficients, we get (6)^2 - 4*(-2)*1=36 + 8=44.

Choose a Method Based on Discriminant

44 is not a perfect square. Therefore the square root method would not provide a simple solution - the quadratic formula must be used instead.

Solve Using Quadratic Formula

The quadratic formula is \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. Substitute the coefficients into the formula to find the solutions for x. \[x = \frac{-6 \pm \sqrt{44}}{-4}\]. This simplifies to \[x_1=\frac{3 - \sqrt{11}}{2}\] and \[x_2=\frac{3 + \sqrt{11}}{2}\]

Check the Solutions

Substitute the solutions back into the original equation to check if they are correct.