The problem is to solve the quadratic equation $x(x+10) = -43$ using the quadratic formula.

1. Problem Description

The problem is to solve the quadratic equation $x(x+10) = -43$ using the quadratic formula.

2. Solution Steps

First, we rewrite the equation in the standard quadratic form $ax^2 + bx + c = 0$.

$x(x+10) = -43$

$x^2 + 10x = -43$

$x^2 + 10x + 43 = 0$

Now, we identify the coefficients: $a = 1$, $b = 10$, and $c = 43$.

We use the quadratic formula to find the solutions for $x$:

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

Substitute the values of $a$, $b$, and $c$ into the quadratic formula:

$x = \frac{-10 \pm \sqrt{10^2 - 4(1)(43)}}{2(1)}$

$x = \frac{-10 \pm \sqrt{100 - 172}}{2}$

$x = \frac{-10 \pm \sqrt{-72}}{2}$

Since the discriminant (the value inside the square root) is negative, the solutions will be complex numbers.

We can simplify the square root of $-72$ as follows:

$\sqrt{-72} = \sqrt{72} \cdot \sqrt{-1} = \sqrt{36 \cdot 2} \cdot i = 6\sqrt{2}i$

Now, substitute this back into the formula for $x$:

$x = \frac{-10 \pm 6\sqrt{2}i}{2}$

Divide both terms in the numerator by 2:

$x = -5 \pm 3\sqrt{2}i$

So, the two solutions are $x = -5 + 3\sqrt{2}i$ and $x = -5 - 3\sqrt{2}i$.

3. Final Answer

$x = -5 + 3\sqrt{2}i, -5 - 3\sqrt{2}i$