Complete the table below to solve the equation 2.5x − 10.5 = 64(0.5^x).

1 answer:

8 0

If you would like to solve the equation 2.5 * x - 10.5 = 64 * 0.5 * x, you can calculate this using the following steps:

<span>2.5 * x - 10.5 = 64 * 0.5 * x
</span>2.5 * x - 10.5 = 32 * x
- 10.5 = 32 * x - 2.5 * x
- 10.5 = 29.5 * x
x = - 10.5 / 29.5
x = - 0.3559 = - 0.36

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Answer:

x is equal to 6

Step-by-step explanation:

4x=24

divide both sides by 4

4 eliminates 4 leaving x behind

4 divides 24 making x =6

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Recall the binomial theorem.

$(a+b)^n = \displaystyle \sum_{k=0}^n \binom nk a^{n-k} b^k$

1. The binomial expansion of $\left(1+\frac x3\right)^7$ is

$\left(1 + \dfrac x3\right)^7 = \displaystyle\sum_{k=0}^7 \binom 7k 1^{7-k} \left(\frac x3\right)^k = \sum_{k=0}^7 \binom 7k \frac{x^k}{3^k}$

Observe that

$k = 1 \implies \dbinom 71 \left(\dfrac x3\right)^1 = \dfrac73 x$

$k = 2 \implies \dbinom 72 \left(\dfrac x3\right)^2 = \dfrac73 x^2$

When we multiply these by $8-9x$ ,

• $8$ and $\frac73 x^2$ combine to make $\frac{56}3 x^2$

• $-9x$ and $\frac73 x$ combine to make $-\frac{63}3 x^2 = -21x^2$

and the sum of these terms is

$\dfrac{56}3 x^2 - 21x^2 = \boxed{-\dfrac73 x^2}$

2. The binomial expansion is

$\left(2a - \dfrac b2\right)^8 = \displaystyle \sum_{k=0}^8 \binom 8k (2a)^{8-k} \left(-\frac b2\right)^k = \sum_{k=0}^8 \binom 8k 2^{8-2k} a^{8-k} b^k$