Solve the equation $2 = x^2 + 10x + 21$, giving your answer to correct 2 decimal places.

Steps to Solve

1. Rewrite the equation Rewrite the given equation $2 = x^2 + 10x + 21$ into the standard quadratic form $ax^2 + bx + c = 0$. Subtract 2 from both sides of the equation: $$x^2 + 10x + 21 - 2 = 0$$ $$x^2 + 10x + 19 = 0$$

2. Apply the quadratic formula Use the quadratic formula to solve for $x$: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ In this case, $a = 1$, $b = 10$, and $c = 19$. Substitute these values into the quadratic formula:

$$x = \frac{-10 \pm \sqrt{10^2 - 4(1)(19)}}{2(1)}$$

3. Simplify the expression Simplify the expression inside the square root: $$x = \frac{-10 \pm \sqrt{100 - 76}}{2}$$ $$x = \frac{-10 \pm \sqrt{24}}{2}$$

4. Further simplification Simplify the square root: $\sqrt{24} = \sqrt{4 \cdot 6} = 2\sqrt{6}$

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

5. Divide by 2 Divide both terms in the numerator by 2:

$$x = -5 \pm \sqrt{6}$$

6. Calculate the two possible values of x Calculate the two possible values of $x$:

$x_1 = -5 + \sqrt{6}$ and $x_2 = -5 - \sqrt{6}$

7. Approximate values Approximate the value of $\sqrt{6} \approx 2.44948974278$. Then: $$x_1 = -5 + 2.44948974278 \approx -2.55051025724$$ $$x_2 = -5 - 2.44948974278 \approx -7.44948974278$$

8. Round to two decimal places Round the values to two decimal places:

$x_1 \approx -2.55$ $x_2 \approx -7.45$