After completing the square.

Question

After completing the square, what are the solutions to the quadratic equation below? $$x^2 + 2x = 25$$

Honstely I think it's B. But I'm not sure.

Answer 1

$$25=x^2+2x=(x+1)^2-1\implies (x+1)^2=26\implies x+1=\pm\sqrt{26}$$

Answer 2

One can got through the process of completing the square (as in "Timbuc"'s posted answer). But one could also check by substitution. So suppose it is proposed that $x=-1+\sqrt{26}$ is a solution. We would then have \begin{align} x^2 + 2x & = (-1+\sqrt{26})^2 + 2(-1+\sqrt{26}) \\[10pt] & = (1-2\sqrt{26}+26) + 2(-1+\sqrt{26}) \\[10pt] & = (1+26)-2 \\[10pt] & = 25, \end{align} so that is indeed a solution. And the same thing works with $-1-\sqrt{26}$.

Answer 3

For the quadratic equation $x^2+bx+c=0$, the sum of the roots is $-b$ and the product is $c$. So for $x^2+2x-25=0$, the $2$ roots sum to $-2$, which eliminates A and B. The product is $-25$. The product of the roots for D is clearly $-24$, which eliminates this answer, leaving C. And it can be quickly verified

$$(-1+\sqrt{26})(-1-\sqrt{26})=(-1)^2-\sqrt{26}^2=-25$$

Answer 4

You can verify your answrt using this. The solution to general quadratic equation $ax^2+bx+c=0$ is given by the formula $\frac {-b+\sqrt {b^2-4ac}}{2a}, \frac {-b-\sqrt {b^2-4ac}}{2a}$ The given equation is $x^2-2x-25=0$ There fore after applying this formula you will get option c as your answer.