Directions "rewrite the standard form equations into slope intercept form" 2x+6y=-24

What are the solutions of the equation x6 + 6x3 + 5 = 0? Use factoring to solve.

lapo4ka [179]

Consider the equation $x^6+6x^3+5=0$ .

First, you can use the substitution $t=x^3$ , then $x^6=(x^3)^2=t^2$ and equation becomes $t^2+6t+5=0$ . This equation is quadratic, so

$D=6^2-4\cdot 5\cdot 1=36-20=16=4^2,\ \sqrt{D}=4,\\ \\ t_{1,2}=\dfrac{-6\pm 4}{2} =-5,-1$ .

Then you can factor this equation:

$(t+5)(t+1)=0$ .

Use the made substitution again:

$(x^3+5)(x^3+1)=0$ .

You have in each brackets the expression like $a^3+b^3$ that is equal to $(a+b)(a^2-ab+b^2)$ . Thus,

$x^3+5=(x+\sqrt[3]{5})(x^2-\sqrt[3]{5}x+\sqrt[3]{25}) ,\\x^3+1=(x+1)(x^2-x+1)$

and the equation is

$(x+\sqrt[3]{5})(x^2-\sqrt[3]{5}x+\sqrt[3]{25})(x+1)(x^2-x+1)=0$ .

Here $x_1=-\sqrt[3]{5} , x_2=-1$ and you can sheck whether quadratic trinomials have real roots:

1. $D_1=(-\sqrt[3]{5}) ^2-4\cdot \sqrt[3]{25}=\sqrt[3]{25} -4\sqrt[3]{25} =-3\sqrt[3]{25}$ .

2. $D_2=(-1)^2-4\cdot 1=1-4=-3$ .

This means that quadratic trinomials don't have real roots.

Answer: $x_1=-\sqrt[3]{5} , x_2=-1$

If you need complex roots, then

$x_{3,4}=\dfrac{\sqrt[3]{5}\pm i\sqrt{3\sqrt[3]{25}}}{2} ,\\ \\x_{5,6}=\dfrac{1\pm i\sqrt{3}}{2}$ .

6 0

3 years ago

The sample mean is x = 12.8%. let x be a random variable that represents the percentage of wheat crop in that county lost to hai

katen-ka-za [31]

Given that $\bar{x}=12.8\%$ and $\sigma=5.0\%$ , given that the population mean is $\mu=11\%$ and the sample size is n = 16, the test statistic is given by

$z= \frac{\bar{x}-\mu}{\sigma/\sqrt{n}} \\ \\ = \frac{12.8-11}{5/\sqrt{16}} \\ \\ = \frac{1.8}{5/4} = \frac{1.8}{1.25} \\ \\ =1.44 \\ \\ \Rightarrow p-value=p(z\ \textgreater \ 1.44) \\ \\ =1-P(z\ \textless \ 1.44)=1-0.92507 \\ \\ =0.07493$

Since the test is a two tailed test, the p-value is 2(0.07493) = 0.14986

Since the p-value is greater the significant level of α = 0.01, thus, we fail to reject the null hypothesis and conclude that there is no sufficient evidence that <span>the percentage of wheat crop lost to hail in that county is different from the national mean of 11%</span>.

8 0

2 years ago

Help....................

leonid [27]

Legs with "B" in their title can not be opposite to the angle B, as they are touching it. So...

The opposite leg to the angle B is the leg AC.

4 0

2 years ago

What is the solution to the system of equations that is graphed below?

iren [92.7K]

Answer:

5 - -3..... i think

Step-by-step explanation:

8 0

2 years ago

F(x)=25x^2-10x+1 what is the the value of the discriminant of f

Elanso [62]

Answer:

0

Step-by-step explanation:

Quadratic equation

$x=\frac{-b+-\sqrt{b^{2}-4*a*c} }{2*a}$

The discriminant is the part of the quadratic formula within the square root symbol: ${b^{2}-4*a*c}$ . The discriminant indicates if there are two solutions, one solution, or none.

The discriminant can be positive, zero or negative which determines how roots exist for the given quadratic equation.

So, a positive discriminant tell us that the quadratic has two different real solutions.

A discriminant of zero tell us that the quadratic has two real and equal solutions.

And a negative discriminant tell us that none of the solutions are real numbers.

In this case: 25x^2-10x+1=0

We can see that

a= 25 b=-10 c=1

Using: ${b^{2}-4*a*c}$

We have $-10^{2}-4* 25*1 =100-100=0$

the answer is zero, so the quadratic has two real and equal solutions

7 0

2 years ago