Completing the Square Flashcards

All 5 cards in this deck

How do you write $x^2+bx+c$ in completed square form?

e.g. $x^2+6x+2$

$\left(x+\frac{b}{2}\right)^2-\left(\frac{b}{2}\right)^2+c$

e.g. $(x+3)^2-9+2=(x+3)^2-7$

What are the steps to solve a quadratic equation by completing the square?

e.g. solve $x^2+6x+2=0$, giving exact answers

1. Write in completed square form: $(x+3)^2-7=0$
2. Rearrange and take both square roots: $x+3=\pm\sqrt{7}$
3. Solve for $x$: $x=-3\pm\sqrt{7}$

True or false? $(x+4)^2=25$ gives only the solution $x=1$.

False. Both the positive and negative square roots are needed: $x+4=\pm5$, so $x=1$ or $x=-9$.

What must you do first to complete the square when $a\neq1$?

e.g. $2x^2+8x+3=0$

Divide every term by $a$ so the coefficient of $x^2$ is $1$.

e.g. $x^2+4x+1.5=0$

What are the solutions of $(x+p)^2=q$?

e.g. $(x-2)^2=5$

$x=-p\pm\sqrt{q}$

e.g. $x=2\pm\sqrt{5}$