Prove that 2x^2+x+8>0 for all real values of x.

Answers

Answer 1

Answer: $y = 2x^2 + x + 8$ is a polynomial of even degree, with a positive $x^2$ coefficient, meaning that
- it will have exactly one turning point
- that turning point will be a minimum

So, if the y-coordinate of the turning point is positive, then this polynomial will be positive for all real values of x.

At a turning point, the gradient of y will be equal to 0. The gradient of y is given by
$\frac{\mathrm{d}y}{\mathrm{d}x} = 4x + 1$ .

To find the turning point, set this equal to 0 and solve for x:
$4x + 1 = 0 \implies x = -(1)/(4)$ .

Substituting this value into the equation gives
$y = 2(-(1)/(4))^2 + (-(1)/(4)) + 8 = (1)/(4) - (1)/(4) + 8 = 8 \ \textgreater \ 0$ .

Since the minimum point of the equation is greater than 0, the equation will be greater than 0 for all real values of x.

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Find the possible values for s in the inequality 12s – 20 ≤ 50 – 3s – 25.

Answers

12s - 20 < 50 - 3s - 25

12s - 20 < 50 - 3s - 25
+3s +3s
15s - 20 < 50 - 25
+ 20 + 20
15s < 70 - 25

15s < 45
÷ 15 ÷15
s < 3

The value of s is less than or equal to 3. So s can be 3, 2, 1, 0, and negative numbers.

s = 3 ⇒ 12(3) - 20 < 50 - 3(3) - 25 ; 36 - 20 < 50 - 9 - 25 ; 16 < 16
s = 2 ⇒ 12(2) - 20 < 50 - 3(2) - 25 ; 24 - 20 < 50 - 6 - 25 ; 4 < 19

Solve for x: ( sqrt 7) ^ 6x = 49 ^ (x-6)

Answers

$(√(7))^(6x)=49^(x-6) \n(7^(1)/(2))^(6x)=(7^2)^(x-6) \n7^{(1)/(2) * 6x}=7^(2(x-6)) \n7^(3x)=7^(2x-12) \n3x=2x-12 \n3x-2x=-12 \nx=-12$

The solution of x for the equation (√7)ˣ = 49ˣ⁻⁶ is,

⇒ x = 8

We have to given that,

An expression to simplify,

⇒ (√7)ˣ = 49ˣ⁻⁶

Now, We can simplify the expression for x as,

⇒ (√7)ˣ = 49ˣ⁻⁶

Since, 49 = 7² = (√7)⁴

Hence,

⇒ (√7)ˣ = (√7)⁴)ˣ⁻⁶

Apply the multiply rule in exponent,

⇒ (√7)ˣ = (√7)⁴ˣ⁻²⁴

By comparing,

⇒ x = 4x - 24

Solve for x,

⇒ x - 4x = - 24

⇒ - 3x = - 24

⇒ 3x = 24

⇒ x = 24 / 3

⇒ x = 8

Therefore, The solution of x for the equation (√7)ˣ = 49ˣ⁻⁶ is,

⇒ x = 8

Learn more about the equation visit:

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HELP!!!! Divide. Write your answer in lowest terms as a proper or improper fraction (not a mixed number). 11/21 divided by 22/7

Answers

My answer is 1/6. Here's How I worked it out:

11/21 / 22/7 is the same as or = to 11/21 * 7/22 which = 77/462 and then divide both the numerator and denominator by 77 and you get 1/6. (In other words simplify 77/462). Hope this helped :)

1/6(6-15d)<2/3(12d+15)
Solve

Answers

      ¹/₆(-15d + 6) < ²/₃(12d + 15)
¹/₆(-15d) + ¹/₆(6) < ²/₃(12d) + ²/₃(15)
      -1¹/₂d + 1 < 8d + 10
      + 1¹/₂d + 1¹/₂d
   1 < 9¹/₂d + 10
   - 10    - 10
      -9 < 9¹/₂d
   ²/₁₉(-9) < ²/₁₉(9¹/₂d)
   ⁻¹⁸/₁₉ < d
   d > ⁻¹⁸/₁₉

Solution Set: {x ∈ x > ⁻¹⁸/₁₉}, (⁻¹⁸/₁₈, ∞)

If s(x) = x – 7 and t(x) = 4x2 – x + 3, which expression is equivalent to (t*s)(x)?4(x – 7)2 – x – 7 + 3

4(x – 7)2 – (x – 7) + 3

(4x2 – x + 3) – 7

(4x2 – x + 3)(x – 7)

Answers

The correct answer is 4(x-7)²-(x-7)+3.

Explanation:
When we find (t*s)(x), that is the same as t(s(x)). To evaluate this, we take the value of s(x), x-7, and substitute it in for every x in t(x). Instead of 4x², we have 4(x-7)²; instead of -x, we have -(x-7). This gives us 4(x-7)²-(x-7)+3.

Answer:

$(4x^2-x+3)(x-7)<strong>$