How do I solve this equation by completing the square?

I'm really confused because there are just variables..please help!

ax^2 + bx + c= 0

Update:

yall are the sweetest, thanks boos xoxo

3 Answers

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  • 8 years ago

    Your teacher is trying to see if you understand how completing the square works - by not using actual numbers. PLUS when we've done it, there's a bonus...

    Just follow the normal steps:

    1) Move the "number" over to the right

    ax^2 + bx = -c

    2) Divide by the "number" multiplying the x^2 (the coefficient)

    x^2 +(b/a) x = -c/a

    3) Take half of the coefficient (don't forget the sign!) of the x-term, and square it. Add this square to both sides of the equation

    The x is multiplied by b/a, so half of that is (b/2a). Squaring it, we get b^2/(4a^2)

    x^2 x^2 +(b/a) x + b^2/(4a^2) = b^2/(4a^2)-c/a

    4) Convert the left to a square

    (x+b/(2a))^2 = b^2/(4a^2)-c/a

    5) Take the square root of both sides

    x + b/(2a) = ±√[b^2/(4a^2)-c/a]

    6) Lets put the left hand side over a common denominator of 4a^2

    x + b/(2a) = ±√[b^2-4ac/4a^2]

    Now we can take that out of the square root

    x + b/(2a) = ±√[b^2-4ac]/2a

    x = -b/(2a)±√[b^2-4ac]/2a

    x = [-b ±√[b^2-4ac]/(2a)

    Recognize it ? Its the quadratic formula!

  • Philo
    Lv 7
    8 years ago

    set c off to the right side out of the way, and factor out a.

    a(x² + b/a x + ....) = -c

    now add half of the x coefficient, squared, inside the parentheses,

    and an equal amount to the right side.

    a[ x² + (b/a)x + (b / 2a)² ] = -c + a( b / 2a)²

    a[ x² + b/a x + b² / 4a² ] = -c + ab² / 4a²

    and rewrite left as a squared binomial while simplifying right

    a[ x + b / 2a]² = -c + b² / 4a

    give right a common denominator and add

    a[ x + b / 2a]² = (-4ac + b²) / 4a

    divide by a to isolate the square on left

    (x + b / 2a)² = (b² – 4ac) / 4a²

    square root each side

    x + b / 2a = ±√(b² – 4ac) / 2a

    and subtract to isolate x

    x = -b / 2a ± √(b² – 4ac) / 2a

  • 8 years ago

    Divide everything by the coefficient of the first term.

    x² + (b/a)x + (c/a) = 0

    Subtract the third term from both sides.

    x² + (b/a)x = -(c/a)

    Add half of the second term coefficient squared to both sides.

    x² + (b/a)x + (b/2a)² = -c/a + (b/2a)²

    That is completing the square, and it leaves the left-hand side with this:

    (x + b/2a)² = -c/a + b²/4a²

    Multiply the first term on the right-hand side by 4a/4a for a common denominator, then simplify.

    (x + b/2a)² = -4ac/4a² + b²/4a²

    (x + b/2a)² = (b² - 4ac) / 4a²

    Take the square root of both sides.

    x + b/2a = ±√(b² - 4ac) / 2a

    Now, subtract b/2a from both sides, giving you the Quadratic Formula

    x = [-b ± √(b² - 4ac)] / 2a

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