Video: Solving Quadratic Equations by Completing Squares

By completing the square, solve the equation 𝑥² + 𝑥 + 1 = 0.

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Video Transcript

By completing the square, solve the equation 𝑥 squared plus 𝑥 plus one is equal to zero.

We recall that any quadratic of the form 𝑥 squared plus 𝑏𝑥 plus 𝑐 can be rewritten as 𝑥 plus 𝑏 over two all squared minus 𝑏 over two squared plus 𝑐. This method is known as completing the square. In this question, the coefficient of 𝑥 and constant term are both equal to one; therefore, 𝑏 equals one and 𝑐 equals one. The expression 𝑥 squared plus 𝑥 plus one can therefore be rewritten as 𝑥 plus a half all squared minus one-half squared plus one. One-half multiplied by one-half is a quarter. Therefore, the equation becomes 𝑥 plus a half all squared minus a quarter plus one is equal to zero. This, in turn, simplifies to 𝑥 plus a half all squared plus three-quarters is equal to zero.

Our next step is to subtract three-quarters from both sides. We can then square root both sides of our equation such that 𝑥 plus a half is equal to the positive or negative square root of negative three-quarters. Square rooting negative three-quarters is the same as square rooting three-quarters and multiplying this by the square root of negative one. From our knowledge of complex numbers, we know that the square root of negative one is equal to 𝑖. The square root of three-quarters is the same as the square root of three over the square root of four, which simplifies to the square root of three over two. 𝑥 plus a half is equal to the positive or negative of root three over two 𝑖.

Finally, we can subtract one-half from both sides of this equation. 𝑥 is equal to negative one-half plus or minus root three over two 𝑖. This means that there are two solutions to the quadratic equation 𝑥 squared plus 𝑥 plus one equals zero. They are 𝑥 is equal to negative a half plus root three over two 𝑖 or 𝑥 is equal to negative a half minus root three over two 𝑖.

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