What value completes the square for the expression? x2 + 14x

Use the quadratic formula to solve the equation. x2 – 7x – 6 = 0

Simplify the expression. (–2i)(8i)

What are the solutions?
1/2x^2 + 2x + 3 =0

I need help

for number one we see that this could be of the form (x+7)^2 so we expand to get x^2+14x+49 so we need to add 49

for number 2 we can just use the easy way (factoring) although you can use the quad formula. (x-1)(x-6) to get x=1,6

we multiply coefficients pretending i is a variable. so we get -16 i^2. we know i is the square root of -1 so i^2 = -1. Thus -16 i^2 = 16.

for the last one, multiply both sides by 2x^2 to get 1+2x^3+6x^2. Then we can let y=x^2. Thus we get 1+2y^2+6y and we rearrange terms to get 2y^2+6y+1.

through the quadratic formula etc. we get a complicated non-real answer: x= (-4+2i\sqrt2)/2, (-4-2i\sqrt2)/2

To complete the square for the expression x^2 + 14x, you can follow these steps:

Step 1: Take the coefficient of x, which is 14, and divide it by 2 to get 7.
Step 2: Square the result from Step 1. 7^2 = 49.
Step 3: Add the result from Step 2 to the expression. x^2 + 14x + 49.

So, the expression x^2 + 14x can be completed as (x + 7)^2.

To solve the equation x^2 – 7x – 6 = 0 using the quadratic formula, you can follow these steps:

Step 1: Identify the coefficients a, b, and c from the equation. In this case, a = 1, b = -7, and c = -6.
Step 2: Use the quadratic formula: x = (-b ± sqrt(b^2 – 4ac)) / (2a).
Step 3: Substitute the values of a, b, and c into the quadratic formula: x = (7 ± sqrt((-7)^2 – 4 * 1 * -6)) / (2 * 1).
Step 4: Simplify the expression inside the square root: x = (7 ± sqrt(49 + 24)) / 2.
Step 5: Further simplify the expression inside the square root: x = (7 ± sqrt(73)) / 2.

So, the solutions of the equation x^2 – 7x – 6 = 0 are x = (7 + sqrt(73)) / 2 and x = (7 - sqrt(73)) / 2.

For the expression (-2i)(8i), you can use the rule that the square root of -1 is represented as "i." Then, simplifying the expression becomes (-2i)(8i) = -16i^2.

Since i^2 represents -1 (by definition), you can replace i^2 with -1 to simplify the expression. So, -16i^2 = -16(-1) = 16.

Therefore, the simplified expression (-2i)(8i) is equal to 16.

For the equation 1/2x^2 + 2x + 3 = 0, you can solve it using various methods. One way to solve it is by factoring or using the quadratic formula.

To solve it using the quadratic formula, you can follow these steps:

Step 1: Identify the coefficients a, b, and c from the equation. In this case, a = 1/2, b = 2, and c = 3.
Step 2: Use the quadratic formula: x = (-b ± sqrt(b^2 – 4ac)) / (2a).
Step 3: Substitute the values of a, b, and c into the quadratic formula: x = (-(2) ± sqrt((2)^2 – 4 * (1/2) * (3))) / (2 * (1/2)).
Step 4: Simplify the expression inside the square root: x = (-2 ± sqrt(4 – 6)) / 1.
Step 5: Further simplify the expression inside the square root: x = (-2 ± sqrt(-2)) / 1.

At this point, the term sqrt(-2) cannot be simplified further using real numbers. However, it is possible to express it using imaginary numbers. Therefore, the solutions will involve imaginary numbers.

If you need further assistance, feel free to ask!