\( \Rightarrow 5 x ^ { 2 } - 45 x \quad 190 = 0 \)

Let's first rewrite the equation clearly: \[ 5x^2 - 45x + 190 = 0 \] Now, we can solve this quadratic equation using the quadratic formula, which is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] In our equation, the coefficients are: - \( a = 5 \) - \( b = -45 \) - \( c = 190 \) Now, let's calculate the discriminant \( b^2 - 4ac \) and then use the quadratic formula to find the values of \( x \). First, we will calculate the discriminant: \[ b^2 - 4ac = (-45)^2 - 4 \cdot 5 \cdot 190 \] Now, I will perform the calculations for the discriminant and then use it to find the roots of the equation. Calculate the value by following steps: - step0: Calculate: \(\left(-45\right)^{2}-4\times 5\times 190\) - step1: Multiply the terms: \(\left(-45\right)^{2}-3800\) - step2: Simplify: \(45^{2}-3800\) - step3: Evaluate the power: \(2025-3800\) - step4: Subtract the numbers: \(-1775\) The discriminant is calculated as follows: \[ b^2 - 4ac = 2025 - 3800 = -1775 \] Since the discriminant is negative (\(-1775\)), this means that the quadratic equation has no real solutions; instead, it has two complex solutions. Now, we can proceed to find the complex solutions using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Substituting the values we have: \[ x = \frac{-(-45) \pm \sqrt{-1775}}{2 \cdot 5} \] This simplifies to: \[ x = \frac{45 \pm \sqrt{-1775}}{10} \] Next, we can express \(\sqrt{-1775}\) in terms of imaginary numbers: \[ \sqrt{-1775} = i\sqrt{1775} \] Now, let's simplify \(\sqrt{1775}\) to find the final solutions for \(x\). Calculate the value by following steps: - step0: Calculate: \(\sqrt{1775}\) - step1: Write as a product: \(\sqrt{25\times 71}\) - step2: Write in exponential form: \(\sqrt{5^{2}\times 71}\) - step3: Use the properties of radicals: \(\sqrt{5^{2}}\times \sqrt{71}\) - step4: Simplify the root: \(5\sqrt{71}\) The square root of \(1775\) simplifies to: \[ \sqrt{1775} = 5\sqrt{71} \] Now, substituting this back into our expression for \(x\): \[ x = \frac{45 \pm i \cdot 5\sqrt{71}}{10} \] We can simplify this further: \[ x = \frac{45}{10} \pm \frac{5i\sqrt{71}}{10} \] This simplifies to: \[ x = 4.5 \pm 0.5i\sqrt{71} \] Thus, the two complex solutions for the equation \(5x^2 - 45x + 190 = 0\) are: \[ x = 4.5 + 0.5i\sqrt{71} \quad \text{and} \quad x = 4.5 - 0.5i\sqrt{71} \]