Question

find all solutions of the equation $x^2 + 10x + 70 = 0$ by using the quadratic formula.
$x = $
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Explanation:

Step1: Recall Quadratic Formula

For a quadratic equation \(ax^2 + bx + c = 0\), the solutions are given by \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\).
Here, \(a = 1\), \(b = 10\), \(c = 70\).

Step2: Calculate Discriminant

Discriminant \(D=b^2 - 4ac=(10)^2-4\times1\times70 = 100 - 280=- 180\).

Step3: Substitute into Formula

\(x=\frac{-10\pm\sqrt{-180}}{2\times1}=\frac{-10\pm\sqrt{180}i}{2}\) (since \(\sqrt{-1}=i\)).
Simplify \(\sqrt{180}=\sqrt{36\times5} = 6\sqrt{5}\).
So \(x=\frac{-10\pm6\sqrt{5}i}{2}=-5\pm3\sqrt{5}i\).

Answer:

\(x=-5 + 3\sqrt{5}i\) or \(x=-5 - 3\sqrt{5}i\)