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3 months ago

What is the solution of the equation (x – 5)2 + 3(x – 5) + 9 = 0? Use u substitution and the quadratic formula to solve.

Mathematics

2 answers:

PIT_PIT [12.4K]3 months ago

8 0

Answer:

x = 1/5 (25 + 3 i sqrt(5)) or x = (-3 i sqrt(5) + 25)/5

Detailed solution using the quadratic formula:

We begin by solving for x:

5 (x - 5)^2 + 9 = 0

Expand the left side:

5 x^2 - 50 x + 134 = 0

Using the quadratic formula:

x = (50 ± sqrt((-50)^2 - 4×5×134))/(2×5) = (50 ± sqrt(2500 - 2680))/10 = (50 ± sqrt(-180))/10:

x = (50 + sqrt(-180))/10 or x = (50 - sqrt(-180))/10

Calculating sqrt(-180): sqrt(-1) sqrt(180) = i sqrt(180):

x becomes (50 + i sqrt(180))/10 or (50 - i sqrt(180))/10

For sqrt(180), we compute: sqrt(4×9×5) = sqrt(2^2×3^2×5) = 2×3sqrt(5) = 6 sqrt(5):

x can be expressed as (i×6 sqrt(5) + 50)/10 or (-i×6 sqrt(5) + 50)/10

Factoring out 2 from 50 + 6 i sqrt(5) gives us 2 (3 i sqrt(5) + 25):

x = 1/102 (3 i sqrt(5) + 25) or x = (-6 i sqrt(5) + 50)/10

(2 (3 i sqrt(5) + 25))/10 simplifies to (2 (3 i sqrt(5) + 25))/(2×5) = (3 i sqrt(5) + 25)/5:

x can be either (3 i sqrt(5) + 25)/5 or (-6 i sqrt(5) + 50)/10

Factoring out 2 from 50 - 6 i sqrt(5) yields 2 (-3 i sqrt(5) + 25):

x becomes 1/5 (25 + 3 i sqrt(5)) or 1/102 (-3 i sqrt(5) + 25)

(2 (-3 i sqrt(5) + 25))/10 can be simplified to (2 (-3 i sqrt(5) + 25))/(2×5) = (-3 i sqrt(5) + 25)/5:

Final Answer: x = 1/5 (25 + 3 i sqrt(5)) or x = (-3 i sqrt(5) + 25)/5

AnnZ [12.3K]3 months ago

6 0

Answer:

$\frac{7\pm 3\sqrt{3}i}{2}$

Step-by-step explanation:

To solve: $(x-5)^2 + 3(x -5) + 9 = 0$

Substituting x-5 = u into the original equation provides us with $u^2+3u+9=0$

We will address this equation through the quadratic formula:

In the equation of form $ax^2+bx+c=0$ , the roots are determined by $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Comparing equation $u^2+3u+9=0$ with $ax^2+bx+c=0$ , we find $a=1\,,\,b=3\,,\,c=9$

Thus, the roots are $u=\frac{-3\pm \sqrt{9-36}}{2}=\frac{-3\pm \sqrt{-27}}{2}=\frac{-3\pm 3\sqrt{3}i}{2}$

Transforming u back to x in $u=\frac{-3\pm 3\sqrt{3}i}{2}$ , leads us to

$x-5=\frac{-3\pm 3\sqrt{3}i}{2}\Rightarrow x=5+\left (\frac{-3\pm 3\sqrt{3}i}{2} \right )=\frac{10-3\pm 3\sqrt{3}i}{2}=\frac{7\pm 3\sqrt{3}i}{2}$

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3 months ago

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Answer:

The function decreases across all real numbers x where x < 1.5.

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762,508 expanded form using exponents

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762,508 expressed in expanded form with exponents is:

$7 \times 10^5 + 6 \times 10^4 + 2 \times 10^3 + 5 \times 10^2 +0 \times 10^1 + 8 \times 10^0$

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We need to write 762,508 using expanded notation with exponents.

Expanded form indicates how to represent numbers to highlight the individual digit values.

Displaying 762508 in Expanded format

Let's determine the positional values of each digit:

The concept of place value signifies the worth assigned to each digit based on its location.

For 7, the place value is 700,000

In 762508, the digit 7 is located in the hundred-thousands position.

For 6, the place value is 60,000

In 762508, the digit 6 occupies the ten-thousands position.

For 2, the place value is 2,000

In 762508, 2 is in the thousands position.

For 5, the place value is 500

In 762508, 5 is placed in the hundreds position.

For 0, the place value is 0 x 10 = 0

In 762508, 0 is found in the tens position.

For 8, the place value is 8

In 762508, 8 is positioned in the units place.

Thus, the expanded form is:

700,000 + 60,000 + 2,000 + 500 + 0 + 8

Expressed with exponents, it can be written as:

$7 \times 10^5 + 6 \times 10^4 + 2 \times 10^3 + 5 \times 10^2 +0 \times 10^1 + 8 \times 10^0$

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