Question

Question 140. In the equation 2x^2 - 3x + 4 = 0, what is the sum of the squares of the roots?

Answer 1

Given the quadratic equation:

`2x^2-3x+4=0`

We solve this equation to find the roots. The general solution for a quadratic equation is given by the expression:

`x=\frac{-b\pm\sqrt[]{b^2-4\cdot a\cdot c}}{2a}`

Looking at the equation, we identify:

`\begin{gathered} a=2 \\ b=-3 \\ c=4 \end{gathered}`

Then, using the general solution formula:

`\begin{gathered} x=\frac{-(-3)\pm\sqrt[]{3^2-4\cdot2\cdot4}}{2\cdot2}=\frac{3\pm\sqrt[]{9-32}}{4} \\ \Rightarrow x_1=\frac{3+\sqrt[]{23}i}{4} \\ \Rightarrow x_2=\frac{3-\sqrt[]{23}i}{4} \end{gathered}`

Now, we need to know the sum of the squares of x₁ and x₂. Then:

`\begin{gathered} x^2_1+x^2_2=(1)/(16)(9+2\cdot\sqrt[]{23}\cdot i-23+9-2\cdot\sqrt[]{23}\cdot i-23) \\ \Rightarrow=(1)/(16)(18-46)=-(28)/(16)=-(7)/(4) \end{gathered}`