Solve the following equation:
(x−4)2+3x2=−16x+12
To solve the given equation, follow these steps:
- Step 1: Expand (x−4)2 using the formula (a−b)2=a2−2ab+b2.
Thus, (x−4)2=x2−8x+16.
- Step 2: Substitute the expanded form into the equation:
x2−8x+16+3x2=−16x+12.
- Step 3: Combine like terms on the left-hand side.
This gives 4x2−8x+16=−16x+12.
- Step 4: Rearrange the equation to set it to zero.
Bring all terms to one side: 4x2−8x+16+16x−12=0.
Combine and simplify the terms: 4x2+8x+4=0.
- Step 5: Simplify the equation by dividing each term by 4.
It becomes x2+2x+1=0.
- Step 6: Recognize the equation as a perfect square trinomial.
(x+1)2=0.
- Step 7: Solve by taking the square root of both sides.
The solution is x+1=0, therefore x=−1.
In conclusion, the solution to the equation is x=−1.