Please solve 4x² + 8x + 16 = 0 and show steps. Confused..
First divide both sides by 4 for a simpler equation. That will not change the answer.
x^2 + 2x + 4 = 0
Note that b^2 -4ac is negative. Use the quadratic equation. The roots are complex, not real.
x = [-2 +/-sqrt(-12)]/2
= -1 +/- i sqrt3
where i is the square root of -1.
To solve the equation 4x² + 8x + 16 = 0, we can use the quadratic formula. The quadratic formula states that for an equation in the form of ax² + bx + c = 0, the solutions can be found using the formula:
x = (-b ± sqrt(b² - 4ac)) / (2a)
Now let's apply this formula to solve the equation step by step.
Step 1: Identify the coefficients a, b, and c from the quadratic equation.
In this case, a = 4, b = 8, and c = 16.
Step 2: Plug the values of a, b, and c into the quadratic formula.
x = (-8 ± sqrt(8² - 4 * 4 * 16)) / (2 * 4)
Simplify inside the square root:
x = (-8 ± sqrt(64 - 256)) / 8
x = (-8 ± sqrt(-192)) / 8
Step 3: Simplify the square root using imaginary numbers.
Since the expression inside the square root (-192) is negative, we can rewrite it as √(-1 * 192) = √(-1) * √(192) = i * √192
Step 4: Simplify further.
x = (-8 ± i√192) / 8
Step 5: Simplify the expression by factoring out 8 from the numerator.
x = -1 ± (i * √192) / 2
Step 6: Simplify √192 further.
To simplify √192, we can break it down into prime factors:
192 = 2 * 2 * 2 * 2 * 2 * 3 = 2^5 * 3
Now rewrite the expression:
x = -1 ± (i * √(2^5 * 3)) / 2
Step 7: Simplify inside the square root:
x = -1 ± (i * √(2^4 * 2 * 3)) / 2
Taking out the square root of 2^4, we get:
x = -1 ± (i * 2^2 * √(2 * 3)) / 2
Simplifying further:
x = -1 ± (i * 4 * √(6)) / 2
Step 8: Simplify the expression:
x = -1 ± (2i * √(6))
So the solutions to the quadratic equation are:
x = -1 + 2i√6
x = -1 - 2i√6
These are the two complex solutions to the equation 4x² + 8x + 16 = 0.