To solve the equation x² – 3 – 2x = 7, we can start by rearranging it into standard form.
First, add 3 to both sides of the equation:
x² - 2x - 3 = 7 + 3
Now the equation is:
x² - 2x - 3 = 10
Next, subtract 10 from both sides to set the equation to zero:
x² - 2x - 3 - 10 = 0
This simplifies to:
x² - 2x - 13 = 0
Now we can factor this quadratic equation or use the quadratic formula. In this case, factoring doesn’t work nicely, so we’ll apply the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
Here, a = 1, b = -2, and c = -13. Plugging these values into the formula gives us:
x = (2 ± √((-2)² - 4(1)(-13))) / (2(1))
Calculating under the square root:
√(4 + 52) = √56 = 2√14
Now substituting back gives us:
x = (2 ± 2√14) / 2
Which simplifies to:
x = 1 ± √14
Thus, the solutions for x are:
x = 1 + √14
x = 1 - √14
In conclusion, the answers for the equation are x = 1 + √14 and x = 1 – √14.