Solve (x-4)² + 3x² = -16x + 12: Multiple Squared Terms Equation

Question

Solve the following equation:

(x4)2+3x2=16x+12 (x-4)^2+3x^2=-16x+12

Video Solution

Solution Steps

00:00 Find X
00:03 Use the shortened multiplication formulas
00:21 Substitute appropriate values according to the given data and open the parentheses
00:37 Substitute into our equation
00:52 Arrange the equation so that one side equals 0
01:14 Collect like terms
01:38 Simplify as much as possible
01:51 Identify the coefficients
02:08 Use the root formula
02:20 Substitute appropriate values and solve
02:45 Calculate the squares and products
02:59 Root 0 always equals 0
03:04 When the root equals 0, there will be only one solution to the equation
03:26 And this is the solution to the problem

Step-by-Step Solution

To solve the given equation, follow these steps:

  • Step 1: Expand (x4)2(x - 4)^2 using the formula (ab)2=a22ab+b2(a - b)^2 = a^2 - 2ab + b^2.

Thus, (x4)2=x28x+16(x - 4)^2 = x^2 - 8x + 16.

  • Step 2: Substitute the expanded form into the equation:

x28x+16+3x2=16x+12x^2 - 8x + 16 + 3x^2 = -16x + 12.

  • Step 3: Combine like terms on the left-hand side.

This gives 4x28x+16=16x+124x^2 - 8x + 16 = -16x + 12.

  • Step 4: Rearrange the equation to set it to zero.

Bring all terms to one side: 4x28x+16+16x12=04x^2 - 8x + 16 + 16x - 12 = 0.

Combine and simplify the terms: 4x2+8x+4=04x^2 + 8x + 4 = 0.

  • Step 5: Simplify the equation by dividing each term by 4.

It becomes x2+2x+1=0x^2 + 2x + 1 = 0.

  • Step 6: Recognize the equation as a perfect square trinomial.

(x+1)2=0(x + 1)^2 = 0.

  • Step 7: Solve by taking the square root of both sides.

The solution is x+1=0x + 1 = 0, therefore x=1x = -1.

In conclusion, the solution to the equation is x=1 x = -1 .

Answer

x=1 x=-1