%5E2%2B6(7x%2B2)%3D27.jpeg "(7x+2)^2+6(7x+2)=27")
Solving the Quadratic Equation: (7x+2)^2 + 6(7x+2) = 27
This article will guide you through the steps of solving the quadratic equation (7x+2)^2 + 6(7x+2) = 27.
1. Simplifying the Equation
First, we need to simplify the equation by expanding the square and distributing the 6:
- (7x+2)^2 = (7x+2)(7x+2) = 49x^2 + 28x + 4
- 6(7x+2) = 42x + 12
Now, the equation becomes:
49x^2 + 28x + 4 + 42x + 12 = 27
2. Rearranging the Equation
Next, we need to move all the terms to one side of the equation to get a standard quadratic equation:
49x^2 + 70x + 16 - 27 = 0
Simplifying further:
49x^2 + 70x - 11 = 0
3. Solving the Quadratic Equation
We can now solve this quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Where:
- a = 49
- b = 70
- c = -11
Substituting these values into the quadratic formula, we get:
x = (-70 ± √(70^2 - 4 * 49 * -11)) / (2 * 49)
Simplifying:
x = (-70 ± √(4900 + 2156)) / 98
x = (-70 ± √7056) / 98
x = (-70 ± 84) / 98
This gives us two solutions:
- x = (-70 + 84) / 98 = 14 / 98 = 1 / 7
- x = (-70 - 84) / 98 = -154 / 98 = -11 / 7
4. Conclusion
Therefore, the solutions to the quadratic equation (7x+2)^2 + 6(7x+2) = 27 are x = 1/7 and x = -11/7.