2%E2%88%9210(x2%E2%88%922)%2B21%3D0.jpeg "(x2−2)2−10(x2−2)+21=0")
Solving the Quadratic Equation: (x² - 2)² - 10(x² - 2) + 21 = 0
This equation might look intimidating at first, but we can solve it by using a simple substitution technique.
Step 1: Substitution
Let's substitute y = x² - 2. This will transform the equation into a more familiar quadratic form:
y² - 10y + 21 = 0
Step 2: Solving the Quadratic Equation
Now, we have a standard quadratic equation in terms of 'y'. We can solve this using the quadratic formula:
y = [-b ± √(b² - 4ac)] / 2a
Where:
- a = 1
- b = -10
- c = 21
Plugging these values into the formula:
y = [10 ± √((-10)² - 4 * 1 * 21)] / 2 * 1
y = [10 ± √(100 - 84)] / 2
y = [10 ± √16] / 2
y = [10 ± 4] / 2
This gives us two possible solutions for 'y':
- y1 = 7
- y2 = 3
Step 3: Back Substitution
Now, we need to substitute back 'x² - 2' for 'y' in both solutions:
- x² - 2 = 7
- x² - 2 = 3
Solving for 'x' in both equations:
- x² = 9
- x = ±3
- x² = 5
- x = ±√5
Final Solutions
Therefore, the solutions for the equation (x² - 2)² - 10(x² - 2) + 21 = 0 are:
- x = 3
- x = -3
- x = √5
- x = -√5