Questions: equation 2x^2-5x+7=0.

Transcript text: equation $2 x^{2}-5 x+7=0$.

Solution

Solution Steps

To solve the quadratic equation $2x^2 - 5x + 7 = 0$, we can use the quadratic formula, which is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Here, $a = 2$, $b = -5$, and $c = 7$. We will calculate the discriminant $b^2 - 4ac$ to determine the nature of the roots and then apply the formula to find the solutions.

Step 1: Identify the Coefficients

The given quadratic equation is $2x^2 - 5x + 7 = 0$. We identify the coefficients as follows:

$a = 2$
$b = -5$
$c = 7$

Step 2: Calculate the Discriminant

The discriminant $\Delta$ of a quadratic equation $ax^2 + bx + c = 0$ is given by: \[ \Delta = b^2 - 4ac \] Substituting the values of $a$, $b$, and $c$, we get: \[ \Delta = (-5)^2 - 4 \times 2 \times 7 = 25 - 56 = -31 \]

Step 3: Determine the Nature of the Roots

Since the discriminant $\Delta = -31$ is negative, the quadratic equation has two complex conjugate roots.

Step 4: Calculate the Roots Using the Quadratic Formula

The roots of the quadratic equation are given by the quadratic formula: \[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \] Substituting the values of $a$, $b$, and $\Delta$, we find: \[ x_1 = \frac{-(-5) + \sqrt{-31}}{2 \times 2} = \frac{5 + \sqrt{-31}}{4} \] \[ x_2 = \frac{-(-5) - \sqrt{-31}}{2 \times 2} = \frac{5 - \sqrt{-31}}{4} \]

Step 5: Express the Roots in Standard Form

The roots can be expressed in the form $a + bi$, where $i$ is the imaginary unit: \[ x_1 = 1.25 + 1.3919i \] \[ x_2 = 1.25 - 1.3919i \]