Solve $x^4-3 x^3+4 x^2-3 x+1=0$
Answer (2)
Solve the quartic equation x 4 − 3 x 3 + 4 x 2 − 3 x + 1 = 0 by dividing by x 2 and substituting y = x + f r a c 1 x , which gives roots x = 1 , 1 , f r a c 1 + i s q r t 3 2 , f r a c 1 − i s q r t 3 2 .
State the quadratic formula: For a x 2 + b x + c = 0 , x = f r a c − b p m s q r t b 2 − 4 a c 2 a .
Prove the quadratic formula by completing the square on a x 2 + b x + c = 0 , leading to the same formula.
The solutions to the quartic equation are x = 1 , 1 , f r a c 1 + i s q r t 3 2 , f r a c 1 − i s q r t 3 2 , and the quadratic formula is x = f r a c − b p m s q r t b 2 − 4 a c 2 a .
Explanation
Problem Overview We are tasked with solving a quartic equation and stating and proving the quadratic formula. Let's break this down into manageable parts.
Solving the Quartic Equation First, let's tackle the quartic equation: x 4 − 3 x 3 + 4 x 2 − 3 x + 1 = 0 Notice that the coefficients are symmetric (1, -3, 4, -3, 1). This suggests that we can divide by x 2 (assuming x = 0 ) and make a substitution to simplify the equation.
Rearranging the Quartic Equation Dividing by x 2 , we get: x 2 − 3 x + 4 − x 3 + x 2 1 = 0 Rearranging the terms, we have: ( x 2 + x 2 1 ) − 3 ( x + x 1 ) + 4 = 0
Substitution Now, let's make a substitution. Let y = x + x 1 . Then y 2 = x 2 + 2 + x 2 1 , so x 2 + x 2 1 = y 2 − 2 . Substituting these into our equation, we get: ( y 2 − 2 ) − 3 y + 4 = 0 This simplifies to: y 2 − 3 y + 2 = 0
Solving for y This is a quadratic equation in y . We can factor it as: ( y − 1 ) ( y − 2 ) = 0 So, y = 1 or y = 2 .
Solving for x Now we need to solve for x in each case. If y = 1 , then x + x 1 = 1 , which gives x 2 − x + 1 = 0 . Using the quadratic formula, we find: x = 2 1 ± 1 − 4 = 2 1 ± i 3 If y = 2 , then x + x 1 = 2 , which gives x 2 − 2 x + 1 = 0 . This factors to ( x − 1 ) 2 = 0 , so x = 1 .
Solutions to Quartic Equation Thus, the solutions to the quartic equation are x = 1 , 1 , 2 1 + i 3 , 2 1 − i 3 .
Stating the Quadratic Formula Now, let's state the quadratic formula. For a quadratic equation of the form a x 2 + b x + c = 0 , the solutions are given by: x = 2 a − b ± b 2 − 4 a c
Proving the Quadratic Formula - Step 1 Finally, let's prove the quadratic formula. Starting with a x 2 + b x + c = 0 , we want to solve for x . First, divide by a (assuming a = 0 ): x 2 + a b x + a c = 0
Proving the Quadratic Formula - Step 2 Now, complete the square: x 2 + a b x + ( 2 a b ) 2 = ( 2 a b ) 2 − a c This gives: ( x + 2 a b ) 2 = 4 a 2 b 2 − a c = 4 a 2 b 2 − 4 a c
Proving the Quadratic Formula - Step 3 Take the square root of both sides: x + 2 a b = ± 4 a 2 b 2 − 4 a c = ± 2 a b 2 − 4 a c
Proving the Quadratic Formula - Step 4 Finally, solve for x : x = − 2 a b ± 2 a b 2 − 4 a c = 2 a − b ± b 2 − 4 a c This completes the proof of the quadratic formula.
Final Answer In summary, the solutions to the quartic equation x 4 − 3 x 3 + 4 x 2 − 3 x + 1 = 0 are x = 1 , 1 , 2 1 + i 3 , 2 1 − i 3 . The quadratic formula for a x 2 + b x + c = 0 is x = 2 a − b ± b 2 − 4 a c , and we have provided a proof of this formula by completing the square.
Examples
The quadratic formula is a fundamental tool in physics, engineering, and economics. For instance, in physics, it can be used to calculate the trajectory of a projectile, where the equation of motion is a quadratic function. In engineering, it helps determine the stability of structures. In economics, it can model supply and demand curves to find equilibrium points. Understanding and applying the quadratic formula allows professionals to solve real-world problems involving quadratic relationships, making it an indispensable tool in various fields.
The solutions to the quartic equation x 4 − 3 x 3 + 4 x 2 − 3 x + 1 = 0 are x = 1 (with multiplicity 2), x = 2 1 + i 3 , and x = 2 1 − i 3 .
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