To solve this problem, you will use the distributive property to create an equation that can be rearranged and solved using the quadratic formula.
<h3>Distribute</h3>
Use the distributive property to distribute 3x into the term (x + 6):
![3x(x+6)=-10](https://tex.z-dn.net/?f=3x%28x%2B6%29%3D-10)
![3x^2+18x=-10](https://tex.z-dn.net/?f=3x%5E2%2B18x%3D-10)
<h3>Rearrange</h3>
To create a quadratic equation, add 10 to both sides of the equation:
![3x^2+18x+10=-10+10](https://tex.z-dn.net/?f=3x%5E2%2B18x%2B10%3D-10%2B10)
![3x^2+18x+10=0](https://tex.z-dn.net/?f=3x%5E2%2B18x%2B10%3D0)
<h3>Use the Quadratic Formula</h3>
The quadratic formula is defined as:
![\displaystyle x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20x%20%3D%20%5Cfrac%7B-b%20%5Cpm%20%5Csqrt%7Bb%5E2%20-%204ac%7D%7D%7B2a%7D)
The model of a quadratic equation is defined as ax² + bx + c = 0. This can be related to our equation.
Therefore:
Set up the quadratic formula:
![\displaystyle x=\frac{-18 \pm \sqrt{(18)^2 - 4(3)(10)}}{2(3)}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20x%3D%5Cfrac%7B-18%20%5Cpm%20%5Csqrt%7B%2818%29%5E2%20-%204%283%29%2810%29%7D%7D%7B2%283%29%7D)
Simplify by using BPEMDAS, which is an acronym for the order of operations:
Brackets
Parentheses
Exponents
Multiplication
Division
Addition
Subtraction
Use BPEMDAS:
![\displaystyle x=\frac{-18 \pm \sqrt{324 - 120}}{6}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20x%3D%5Cfrac%7B-18%20%5Cpm%20%5Csqrt%7B324%20-%20120%7D%7D%7B6%7D)
Simplify the radicand:
![\displaystyle x=\frac{-18 \pm \sqrt{204}}{6}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20x%3D%5Cfrac%7B-18%20%5Cpm%20%5Csqrt%7B204%7D%7D%7B6%7D)
Create a factor tree for 204:
204 - 1, 2, 3, 4, 6, 12, 17, 34, 51, 68, 102 and 204.
The largest factor group that creates a perfect square is 4 × 51. Therefore, turn 204 into 4 × 51:
![\sqrt{4\times51}](https://tex.z-dn.net/?f=%5Csqrt%7B4%5Ctimes51%7D)
Then, using the Product Property of Square Roots, break this into two radicands:
![\sqrt{4} \times \sqrt{51}](https://tex.z-dn.net/?f=%5Csqrt%7B4%7D%20%5Ctimes%20%5Csqrt%7B51%7D)
Since 4 is a perfect square, it can be evaluated:
![2 \times \sqrt{51}](https://tex.z-dn.net/?f=2%20%5Ctimes%20%5Csqrt%7B51%7D)
To simplify further for easier reading, remove the multiplication symbol:
![2\sqrt{51}](https://tex.z-dn.net/?f=2%5Csqrt%7B51%7D)
Then, substitute for the quadratic formula:
![\displaystyle x=\frac{-18 \pm 2\sqrt{51}}{6}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20x%3D%5Cfrac%7B-18%20%5Cpm%202%5Csqrt%7B51%7D%7D%7B6%7D)
This gives us a combined root, which we should separate to make things easier on ourselves.
<h3>Separate the Roots</h3>
Separate the roots at the plus-minus symbol:
![\displaystyle x=\frac{-18 + 2\sqrt{51}}{6}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20x%3D%5Cfrac%7B-18%20%2B%202%5Csqrt%7B51%7D%7D%7B6%7D)
![\displaystyle x=\frac{-18 - 2\sqrt{51}}{6}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20x%3D%5Cfrac%7B-18%20-%202%5Csqrt%7B51%7D%7D%7B6%7D)
Then, simplify the numerator of the roots by factoring 2 out:
![\displaystyle x=\frac{2(-9 + \sqrt{51})}{6}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20x%3D%5Cfrac%7B2%28-9%20%2B%20%5Csqrt%7B51%7D%29%7D%7B6%7D)
![\displaystyle x=\frac{2(-9 - \sqrt{51})}{6}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20x%3D%5Cfrac%7B2%28-9%20-%20%5Csqrt%7B51%7D%29%7D%7B6%7D)
Then, simplify the fraction by reducing 2/6 to 1/3:
![\boxed{\displaystyle x=\frac{-9 + \sqrt{51}}{3}}](https://tex.z-dn.net/?f=%5Cboxed%7B%5Cdisplaystyle%20x%3D%5Cfrac%7B-9%20%2B%20%5Csqrt%7B51%7D%7D%7B3%7D%7D)
![\boxed{\displaystyle x=\frac{-9 - \sqrt{51}}{3}}](https://tex.z-dn.net/?f=%5Cboxed%7B%5Cdisplaystyle%20x%3D%5Cfrac%7B-9%20-%20%5Csqrt%7B51%7D%7D%7B3%7D%7D)
The final answer to this problem is:
![\displaystyle x=\frac{-9 - \sqrt{51}}{3}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20x%3D%5Cfrac%7B-9%20-%20%5Csqrt%7B51%7D%7D%7B3%7D)