Solving Quadratic Equations by Completing the Square

A quadratic equation in standard form:
ax² + bx + c = 0
can be rewritten in perfect square form:
(x+d)² = e
where d and e are constants.
We can then take the square root of both sides and then solve for x.

The following diagram shows how to use the Completing the Square method to solve quadratic equations. Scroll down the page for more examples and solutions of solving quadratic equations using completing the square.

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Steps to solve a quadratic equation by completing the square

1. Make the coefficient of x² equal to 1 (if a ≠ 1):
   Divide every term in the equation by a. This will give you an equation of the form:
   \(x^2 + \frac{b}{a}x + \frac{c}{a}\)
   
2. Move the constant term to the right side of the equation:
   Subtract \(\frac{c}{a}\) from both sides:
   \(x^2 + \frac{b}{a}x = -\frac{c}{a}\)
   
3. Complete the square on the left side:
   Take half of the coefficient of the x term to get \(\frac{b}{2a}\)
   Square the value: \((\frac{b}{2a})^2 = \frac{b^2}{4a^2}\)
   Add this squared value to both sides of the equation: \(x^2 + \frac{b}{a}x + \frac{b^2}{4a^2} = -\frac{c}{a} + \frac{b^2}{4a^2}\)
   
4. Factor the left side as a perfect square trinomial:
   The left side will now factor into \((x + \frac{b}{2a})^2\)
   
5. Take the square root of both sides and solve for x.

Completing the Square - Solving Quadratic Equations
Examples:

1. x² + 6x - 7 = 0
2. 2x² - 10x - 3 = 0
3. -x² - 6x + 7 = 0

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Quadratic Equations - Completing the Square
Examples:

1. x² + 12x - 31 = 0
2. x² - 8x + 10 = 0
3. 2x² - 10x + 8 = 0
4. 3x² - 8x + 5 = 0

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Solving a quadratic by completing the square

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