Question

Solve the equation 3x+x²=-7. Does the number of solutions match the information provided by the discriminant? Explain

Answer 1

Answer:

$-\frac{3}{2}+\frac{\sqrt{19}i}{2}$ and $-\frac{3}{2}-\frac{\sqrt{19}i}{2}$

Step-by-step explanation:

$3x+x^2=-7\\\Rightarrow x^2+3x+7=0$

Discriminant

$D=b^2-4ac\\\Rightarrow D=3^2-4\times 7\times 1\\\Rightarrow D=-19$

If the discriminant is a negative number then the roots of the equation are imaginary

$x=\frac{-b\pm \sqrtD}{2a}\\\Rightarrow x=\frac{-3\pm \sqrt{-19}}{2\times 1}\\\Rightarrow x=-\frac{3}{2}\pm \frac{\sqrt{19}i}{2}\\\Rightarrow x=-\frac{3}{2}+\frac{\sqrt{19}i}{2}, x=-\frac{3}{2}-\frac{\sqrt{19}i}{2}$

So, the two solutions are $-\frac{3}{2}+\frac{\sqrt{19}i}{2}$ and $-\frac{3}{2}-\frac{\sqrt{19}i}{2}$