Solve the Quadratic Equation: 2(a-4)² + 3 = 163-16a

Video Solution

Solution Steps

00:00 Find A

00:04 Use shortened multiplication formulas to open the parentheses

00:32 Solve the multiplications and squares

00:44 Open parentheses properly, multiply by each factor

01:03 Simplify what we can

01:14 Isolate A

01:40 Extract the root

01:44 When extracting a root, there are always 2 possibilities: negative and positive

01:47 And this is the solution to the question

Step-by-Step Solution

To solve the given equation $2(a-4)^2 + 3 = 163 - 16a$ , we begin by expanding the expression $(a-4)^2$ .

Step 1: Expand $(a-4)^2$ : $(a-4)^2 = a^2 - 8a + 16$ .
Step 2: Multiply the expanded form by 2: $2(a^2 - 8a + 16) = 2a^2 - 16a + 32$ .
Step 3: Substitute this into the equation and combine like terms: $2a^2 - 16a + 32 + 3 = 163 - 16a$ . This simplifies to $2a^2 - 16a + 35 = 163 - 16a$ .
Step 4: Move all terms to one side of the equation: $2a^2 -16a - 16a + 35 - 163 = 0$ $2a^2 - 32a - 128 = 0$ .
Step 5: Simplify the quadratic equation by dividing by 2: $a^2 - 16a - 64 = 0$ .
Step 6: Solve the quadratic equation by factoring: $a^2 - 16a - 64 = (a - 8)^2 - 64 = (a - 8)(a + 8) = 0$ .

Finding the roots gives $a - 8 = 0$ or $a + 8 = 0$ . Thus, $a = 8$ or $a = -8$ .

Therefore, the solutions to the equation are $\pm 8$ .

Thus, the correct answer is $\pm 8$ .