I need help with Math 1010: Completing the square.

Given equation:
\[ x^2 - 9x = 10 \]

x 2 − 9 x = 10∣ + 20.25 x 2 − 9 x + 20.25 = 30.25 ( x − 4.5 ) 2 = 30.25 x − 4.5 = 5.5 ∨ x − 4.5 = − 5.5 x = 10 ∨ x = − 1

Answered by konrad509 | 2024-06-10

The solutions to the equation x 2 − 9 x = 10 after completing the square are x = 10 and x = − 1 .
To complete the square for the quadratic equation x 2 − 9 x = 10 , follow these steps:

Start with the given quadratic equation:

x 2 − 9 x = 10

Move the constant term to the other side of the equation to prepare for completing the square:

x 2 − 9 x − 10 = 0

To complete the square, take the coefficient of x , which is − 9 , divide it by 2, and square the result to find the value that needs to be added to both sides of the equation:

( 2 − 9 ​ ) 2 = ( 4 81 ​ )

Add this value to both sides of the equation:

x 2 − 9 x + 4 81 ​ = 10 + 4 81 ​

Simplify the right side of the equation:

10 + 4 81 ​ = 4 40 ​ + 4 81 ​ = 4 121 ​

Now the equation is in the form of a perfect square trinomial on the left side:

( x − 2 9 ​ ) 2 = 4 121 ​

Take the square root of both sides to solve for x :

x − 2 9 ​ = ± 4 121 ​ ​

Simplify the square root:

x − 2 9 ​ = ± 2 11 ​

Solve for x by isolating it on one side:

x = 2 9 ​ ± 2 11 ​

Find the two solutions:

x = 2 9 ​ + 2 11 ​ = 2 20 ​ = 10
x = 2 9 ​ − 2 11 ​ = 2 − 2 ​ = − 1
Therefore, the solutions to the equation x 2 − 9 x = 10 after completing the square are x = 10 and x = − 1 .

Answered by MuskanJa | 2024-06-24

To solve the equation x 2 − 9 x = 10 by completing the square, rearrange it, add the necessary value to both sides, and simplify. The final answers are x = 10 and x = − 1 . This method allows you to visualize the quadratic function and find its roots accurately.
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Answered by MuskanJa | 2024-12-26