solve (v+6)^2 -40=0, where v is a real number.
(v+6)^2-40 = 0
(v+6)^2 = 40
v+6 = ±√40
v = -6±√40
or,
v = -6±2√10
( v + 6 )² - 40 = 0
Add 40 to both sides
( v + 6 )² - 40 + 40 = 0 + 40
( v + 6 )² = 40
v + 6 = ± √ 40
v + 6 = ± √4 ∙ √10
v + 6 = ± 2 √10
Subtract 6 to both sides
v + 6 - 6 = ± 2 √10 - 6 =
± 2 ( √10 - 3 )
The solutions are:
v₁ = - 2 √10 - 6 = - 2 ( √10 + 3 )
v₂ = 2 √10 - 6 = 2 ( √10 - 3 )
To solve the equation (v+6)^2 - 40 = 0, we need to eliminate the square term first. Here are the steps to solve the equation:
1. Expand the square term by multiplying (v+6) by itself:
(v+6)(v+6) - 40 = 0.
This gives us v^2 + 12v + 36 - 40 = 0.
2. Simplify the equation:
v^2 + 12v - 4 = 0.
3. Now, we have a quadratic equation in the form of av^2 + bv + c = 0, where a = 1, b = 12, and c = -4.
4. To solve the quadratic equation, we can use either factoring, completing the square, or the quadratic formula. In this case, we will use the quadratic formula:
The quadratic formula is given by:
v = (-b ± √(b^2 - 4ac)) / (2a).
Plugging in the values for a, b, and c, we get:
v = (-(12) ± √((12)^2 - 4(1)(-4))) / (2(1)).
Simplifying further:
v = (-12 ± √(144 + 16)) / 2.
v = (-12 ± √160) / 2.
v = (-12 ± √(16*10)) / 2.
v = (-12 ± 4√10) / 2.
5. We can simplify the expression by dividing the numerator and denominator by 2:
v = -6 ± 2√10.
Hence, the solutions to the equation (v+6)^2 - 40 = 0 are v = -6 + 2√10 and v = -6 - 2√10.