Solve the equation by completing the square 16x^2+1=7x!!! Please help me ASAP!!!! :(
16 x² + 1 = 7 x
Subtract 7 x to both sides
16 x² + 1 - 7 x = 7 x - 7 x
16 x² - 7 x + 1 = 0
Divide both sides by 16
x² - 7 x / 16 + 1 / 16 = 0
Subtract 1 / 16 to both sides
x² - 7 x / 16 + 1 / 16 - 1 / 16 = 0 - 1 / 16
x² - 7 x / 16 = - 1 / 16
Add [ ( 7 / 16 ) / 2 ]² = ( 7 / 32 )² to both sides
x² - 7 x / 16 + [ ( 7 / 32 ) ]² = - 1 / 16 + [ ( 7 / 32 ) ]²
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Remark:
( x - 7 / 32 )² = x² - 2 x ∙ 7 / 32 + ( 7 / 32 )²
( x - 7 / 32 )² = x² - 2 ∙ 7 x / 2 ∙ 16 + ( 7 / 32 )²
( x - 7 / 32 )² = x² - 7 x / 16 + ( 7 / 32 )²
So:
x² - 7 x / 16 + ( 7 / 32 )² = ( x - 7 / 32 )²
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( x - 7 / 32 )² = - 1 / 16 + 49 / 1024
( x - 7 / 32 )² = - 1 ∙ 64 / 16 ∙ 64 + 49 / 1024
( x - 7 / 32 )² = - 64 / 1024 + 49 / 1024
( x - 7 / 32 )² = - 15 / 1024
Take square root of both sides
x - 7 / 32 = ± √ ( - 15 / 1024 )
x - 7 / 32 = ± √ ( - 15 ) / √1024
x - 7 / 32 = ± √ [ ( - 1 ) ∙ √15 ] / 32
x - 7 / 32 = ± √ ( - 1 ) ∙ √15 / 32
x - 7 / 32 = ± i ∙ √15 / 32
Add 7 / 32 to both sides
x - 7 / 32 + 7 / 32 = 7 / 32 ± i ∙ √15 / 32
x = 7 / 32 ± i ∙ √15 / 32
x = ( 1 / 32 ) ∙ ( 7 ± i ∙ √15 )
x1= ( 1 / 32 ) ∙ ( 7 + i ∙ √15 )
x2 = ( 1 / 32 ) ∙ ( 7 - i ∙ √15 )
My typo:
it is written
x - 7 / 32 = ± √ [ ( - 1 ) ∙ √15 ] / 32
should be
x - 7 / 32 = ± √ [ ( - 1 ) ∙ 15 ] / 32
Sure, I can help you solve the equation 16x^2 + 1 = 7x by completing the square. Here's the step-by-step process:
Step 1: Move all terms to one side of the equation to have a quadratic expression equal to zero:
16x^2 - 7x + 1 = 0
Step 2: Divide the entire equation by the leading coefficient (the coefficient of x^2) to make it equal to 1. In this case, divide everything by 16:
x^2 - (7/16)x + 1/16 = 0
Step 3: Next, focus on completing the square for the quadratic expression x^2 - (7/16)x.
Step 4: Take half of the coefficient of x (in this case, -7/16) and square it to get the value to add and subtract to the equation. Half of -7/16 is -7/32, and squaring it gives us (7/32)^2 = 49/1024.
Step 5: Add and subtract the value obtained in step 4 to complete the square:
x^2 - (7/16)x + 49/1024 - 49/1024 + 1/16 = 0
Simplifying, we get:
(x - 7/32)^2 - 49/1024 + 64/1024 = 0
(x - 7/32)^2 + 15/1024 = 0
Step 6: Now that the equation is in the form of (x - h)^2 + k = 0, we can see that the vertex of the equation is at (h, k). In this case, the vertex is (7/32, -15/1024).
Step 7: Since we want to solve for x, we set the expression inside the square (x - 7/32)^2 equal to the negative value of k. In this case, -15/1024:
(x - 7/32)^2 = -15/1024
Step 8: Taking the square root of both sides, remembering to consider both positive and negative roots:
x - 7/32 = ± √(-15/1024)
Step 9: Simplifying the square root:
x - 7/32 = ± i√(15/1024)
Step 10: Isolating x, add 7/32 to both sides:
x = 7/32 ± i√(15/1024)
Therefore, the solutions to the equation 16x^2 + 1 = 7x are:
x = 7/32 + i√(15/1024)
x = 7/32 - i√(15/1024)
Note that both solutions involve complex numbers, as indicated by the presence of "i" in the expressions.