x^2=8x-9
solve by completing the square to obtain exact solutions
x^2 - 8x + .... = -9 + ....
x^2 - 8x + 16 = -9 + 16
(x-4)^2 = 7
x-4 ± √7
x = 4 ± √7
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To solve the equation x^2 = 8x - 9 by completing the square, first make sure the equation is in the form x^2 + bx = c. In this case, we already have x^2 and 8x, but we need to move the constant term (-9) to the other side of the equation:
x^2 - 8x = 9
Now, we can complete the square. To do this, take half of the coefficient of x (which is -8 in this case), square it, and add it to both sides of the equation:
x^2 - 8x + (-8/2)^2 = 9 + (-8/2)^2
Simplifying this equation gives:
x^2 - 8x + 16 = 9 + 16
We can simplify further on both sides:
(x - 4)^2 = 25
Next, take the square root of both sides of the equation, considering both positive and negative roots:
√((x - 4)^2) = ±√25
Simplifying:
x - 4 = ±5
Now, solve for x by adding 4 to both sides of the equation:
x = 4 ± 5
Therefore, the exact solutions of the equation x^2 = 8x - 9 obtained by completing the square are x = 4 + 5 and x = 4 - 5, which can be further simplified as x = 9 and x = -1.
To solve the equation x^2 = 8x - 9 by completing the square, follow these steps:
Step 1: Move the constant term to the right side of the equation:
x^2 - 8x = -9
Step 2: Take half of the coefficient of 'x' (which is -8) and square it:
(-8/2)^2 = (-4)^2 = 16
Step 3: Add this squared value to both sides of the equation:
x^2 - 8x + 16 = -9 + 16
x^2 - 8x + 16 = 7
Step 4: Rewrite the left side of the equation as a perfect square trinomial:
(x - 4)^2 = 7
Step 5: Take the square root of both sides of the equation, remembering to include both the positive and negative square root:
x - 4 = ±√7
Step 6: Solve for 'x' by adding 4 to both sides of the equation:
x = 4 ± √7
Therefore, the exact solutions to the equation x^2 = 8x - 9 obtained by completing the square are x = 4 + √7 and x = 4 - √7.