How to solve the quadratic formula by completing the square

Completing the square So far, you've either solved quadratic equations by taking the square root or by factoring. These methods are relatively simple and efficient, when applicable.

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The quadratic equation in the previous page's last example was:. The expression on the left-hand side of this equation can be multiplied out and simplified to be:. But we still would not have been able to solve the equation, even with the quadratic formatted this way, because it doesn't factor and it isn't ready for square-rooting. Completing the Square. The only reason we could solve it on the previous page was because they'd already put all the x stuff inside a square, so we could move the strictly-numerical portion of the equation to the other side of the "equals" sign and then square-root both sides. They won't always format things as nicely as this. So how do we go from a regular quadratic like the above to an equation that is ready to be square-rooted?

The Process The Formula. Some quadratics are fairly simple to solve because they are of the form "something-with- x squared equals some number", and then you take the square root of both sides. Completing the Square. Unfortunately, most quadratics don't come neatly squared like this. For your average everyday quadratic, you first have to use the technique of "completing the square" to rearrange the quadratic into the neat " squared part equals a number " format demonstrated above. For example:.

If you're seeing this message, it means we're having trouble loading external resources on our website. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Solving quadratics by completing the square. Worked example: Completing the square intro. Practice: Completing the square intro. Worked example: Rewriting expressions by completing the square.

But if you have time, let me show you how to " Complete the Square " yourself. Having x twice in the same expression can make life hard. What can we do? Otherwise the whole value changes. We can complete the square to solve a Quadratic Equation find where it is equal to zero. But a general Quadratic Equation can have a coefficient of a in front of x 2 :.

How Do You Solve a Quadratic Equation by Completing the Square?

Completing the Square is a method used to solve a quadratic equation by changing the form of the equation so that the left side is a perfect square trinomial. Transform the equation so that the constant term, c , is alone on the right side. If a , the leading coefficient the coefficient of the x 2 term , is not equal to 1 , divide both sides by a. Add the square of half the coefficient of the x -term, b 2 a 2 to both sides of the equation. Take the square root of both sides. Solve for x.

Easy to understand algebra lessons on DVD. Try before you commit. For quadratic equations that cannot be solved by factorising, we use a method which can solve ALL quadratic equations called completing the square. We use this later when studying circles in plane analytic geometry. Completing the square comes from considering the special formulas that we met in Square of a sum and square of a difference earlier:. Solving Quadratics by Completing the Square

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Demonstrates, with step-by-step instructions and illustrations, how to complete the square to solve a quadratic equation.

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3 thoughts on “How to solve the quadratic formula by completing the square”

1. Scott F. says:

We can complete the square to solve a Quadratic Equation (find where it is equal to zero). But a general Quadratic Equation can have a coefficient of a in front of.

2. Rosie M. says:

Completing the Square

3. Ane M. says: