Questions: For the quadratic equation 2x^2 - 4x + 5 = 0, find the value of the discriminant to determine if the equation has a real or non-real solution. a.) sqrt(24); real solution b.) sqrt(36); real solution c.) sqrt(-24); non-real solution d.) sqrt(-36); non-real solution

For the quadratic equation 2x^2 - 4x + 5 = 0, find the value of the discriminant to determine if the equation has a real or non-real solution.
a.) sqrt(24); real solution
b.) sqrt(36); real solution
c.) sqrt(-24); non-real solution
d.) sqrt(-36); non-real solution

Transcript text: For the quadratic equation $2 x^{2}-4 x+5=0$, find the value of the discriminant to determine if the equation has a real or non-real solution. a.) $\sqrt{24}$; real solution b.) $\sqrt{36}$; real solution c.) $\sqrt{-24}$; non-real solution d.) $\sqrt{-36}$; non-real solution

Solution

Solution Steps

To determine if the quadratic equation has real or non-real solutions, we need to calculate the discriminant. The discriminant $ D $ of a quadratic equation $ ax^2 + bx + c = 0 $ is given by $ D = b^2 - 4ac $. If $ D > 0 $, the equation has two distinct real solutions. If $ D = 0 $, it has exactly one real solution. If $ D < 0 $, the solutions are non-real (complex).

Step 1: Calculate the Discriminant

For the quadratic equation $2x^2 - 4x + 5 = 0$, we identify the coefficients as $a = 2$, $b = -4$, and $c = 5$. The discriminant $D$ is calculated using the formula: \[ D = b^2 - 4ac \] Substituting the values, we have: \[ D = (-4)^2 - 4 \cdot 2 \cdot 5 = 16 - 40 = -24 \]

Step 2: Determine the Nature of the Solutions

Since the discriminant $D = -24$ is less than zero ($D < 0$), this indicates that the quadratic equation has non-real (complex) solutions.