Question

Please solve with explanation

Answer 1

`\textsf{a)}\quad y=8x^2+2x-3`

`\textsf{b)}\quad y=x^2-2x-4`

Explanation:

Given information:

`x^2+bx+c=0`

`x_1+x_2=-b`

`x_1x_2=c`

Part (a)

`\textsf{If}\quad x_1=\frac12\quad\textsf{and}\quad x_2=-\frac34`

`\begin{aligned}\implies -b & =\frac12+\left(-\frac34\right)\\ & = \frac24-\frac34\\ & =-\frac14\end{aligned}`

`\begin{aligned}\implies c & =\frac12 \cdot \left(-\frac34\right)\\ & = (1 (-3))/(2 \cdot 4)\\ & =-\frac38\end{aligned}`

Substituting the found values of b and c into
`x^2+bx+c=0`

`\implies x^2+\frac14x-\frac38=0`

Multiply both sides by 8 so that coefficients are integers:

`\implies 8x^2+2x-3=0`

Therefore, the final quadratic equation is:

`y=8x^2+2x-3`

----------------------------------------------------------------------------------------------

Part (b)

`\textsf{If}\quad x_1=1+√(5)\quad\textsf{and}\quad x_2=1-√(5)`

`\begin{aligned}\implies -b & =(1+√(5))+(1-√(5))\\ & = 1+1+√(5)-√(5)\\ & =2\end{aligned}`

`\begin{aligned}\implies c & =(1+√(5))(1-√(5))\\ & = 1-√(5)+√(5)-5\\ & =-4\end{aligned}`

Substituting the found values of b and c into
`x^2+bx+c=0`

`\implies x^2-2x-4=0`

Therefore, the final quadratic equation is:

`y=x^2-2x-4`