Roots and coefficient

In the equation 2x ^ 2 + bx-9 = 0 is one root x1 = -3/2. Determine the second root and the coefficient b.

Correct answer:

b =  -3
x₂ =  3

Step-by-step explanation:

$$\begin{gathered} a=-3\mathrm{/}2=-{\displaystyle \frac{3}{2}}=-1\frac{1}{2}=-1.5 \\ 2\cdot a^2+b\cdot a-9=0 \\ 2\cdot (-1.5{)}^2+b\cdot (-1.5)-9=0 \\ 1.5b=-4.5 \\ b=-3 \end{gathered}$$

$$\begin{gathered} 2\ast x^2+b\ast x-9=0 \\ 2\cdot x^2+(-3)\cdot x-9=0 \\ 2x^2-3x-9=0 \\ a=2;b=-3;c=-9 \\ D=b^2-4ac=3^2-4\cdot 2\cdot (-9)=81 \\ D>0 \\ {x}_{1,2}={\displaystyle \frac{-b\pm \sqrt{D}}{2a}}={\displaystyle \frac{3\pm \sqrt{81}}{4}} \\ {x}_{1,2}={\displaystyle \frac{3\pm 9}{4}} \\ {x}_{1,2}=0.75\pm 2.25 \\ {x}_1=3 \\ {x}_2=-1.5 \\ \text{ Factored form of the equation: } \\ 2(x-3)(x+1.5)=0 \\ {x}_2=x_1=3 \end{gathered}$$

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