Question

It’s asking me to use the quadratic formula to solve the roots in the following equation which is in the picture

Answer 1

Solve the equation:

`4x^2+5x+2=2x^2+7x-1`

We need to take all terms of the equation to the left side. We do that by subtracting each one of them from both sides of the equation as follows:

`4x^2+5x+2-2x^2-7x+1=0`

Collecting like terms:

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

Identify the coefficients: a = 2, b = -2, c = 3. And apply the quadratic formula:

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

Substituting:

`x=(-(-2)\pm√((-2)^2-4(2)(3)))/(2(2))`

Operating:

`\begin{gathered} x=(2\pm√(4-24))/(4) \\ \\ x=(2\pm√(-20))/(4) \end{gathered}`

The solutions to the equation are complex because the square root of a negative number is imaginary:

`x=(2\pm√(20)i)/(4)`

Since 20 = 4*5, this expression can be simplified:

`\begin{gathered} x=(2\pm2√(5)i)/(4) \\ \\ x=(1\pm√(5)\imaginaryI)/(2) \end{gathered}`

Finally, separating the real and imaginary parts:

`x=(1)/(2)\pm(√(5))/(2)i`