Algebra 2 Flashcards: Complete The Square To Find Solutions

What this deck covers

This deck focuses on Complete The Square To Find Solutions, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Solve by completing the square: $x^2+2x-7=0$.

Answer: $x=-1\pm 2\sqrt{2}$. From $(x+1)^2=8$, take square root: $x=-1\pm 2\sqrt{2}$.

Flashcard 2: What are the solutions of $(x-p)^2=q$ written explicitly?

Answer: $x=p\pm\sqrt{q}$. Explicit form after isolating $x$.

Flashcard 3: Rewrite $x^2+9x$ as a square plus a constant: x^2+9x=\left(x+__\right)^2-__.

Answer: $\left(x+\frac{9}{2}\right)^2-\frac{81}{4}$. Half of $9$ is $\frac{9}{2}$, then subtract $\left(\frac{9}{2}\right)^2$.

Flashcard 4: Transform $2x^2+8x+3=0$ into $(x-p)^2=q$ form.

Answer: $\left(x+2\right)^2=\frac{5}{2}$. First divide by $2$, then complete the square.

Flashcard 5: What is the quadratic formula for solutions to $ax^2+bx+c=0$?

Answer: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived by completing the square on the general form.

Flashcard 6: What expression is the discriminant in $ax^2+bx+c=0$?

Answer: $b^2-4ac$. Determines the nature of quadratic solutions.

Flashcard 7: What does $b^2-4ac>0$ tell you about the solutions of $ax^2+bx+c=0$?

Answer: Two distinct real solutions. Positive discriminant means two real intersection points.

Flashcard 8: What does $b^2-4ac=0$ tell you about the solutions of $ax^2+bx+c=0$?

Answer: One real double root. Zero discriminant means one repeated real solution.

Flashcard 9: What does $b^2-4ac<0$ tell you about the solutions of $ax^2+bx+c=0$?

Answer: No real solutions (two complex solutions). Negative discriminant means no real intersection points.

Flashcard 10: What is the vertex form obtained by completing the square for $y=ax^2+bx+c$?

Answer: $y=a(x-h)^2+k$. Standard vertex form from completing the square.

Flashcard 11: What is $h$ in vertex form $y=a(x-h)^2+k$ in terms of $a$ and $b$?

Answer: $h=-\frac{b}{2a}$. Formula for the x-coordinate of the vertex.

Flashcard 12: What is the key identity used to expand $(x+p)^2$ while completing the square?

Answer: $(x+p)^2=x^2+2px+p^2$. Fundamental binomial expansion used in completing squares.

Flashcard 13: What operation do you use after writing a quadratic as $(x-p)^2=q$ to solve for $x$?

Answer: Take square roots: $x-p=\pm\sqrt{q}$. Square root both sides to solve for $x$.

Flashcard 14: What is the completing-the-square step after getting $x^2+bx$ on one side?

Answer: Add $\left(\frac{b}{2}\right)^2$ to both sides. Maintains equation balance while creating a perfect square.

Flashcard 15: What is the first step to complete the square in $ax^2+bx+c=0$ when $a\neq 1$?

Answer: Divide by $a$ to make the $x^2$ coefficient $1$. Makes the leading coefficient 1 for easier completion.

Flashcard 16: What perfect square trinomial equals $x^2+bx+\left(\frac{b}{2}\right)^2$?

Answer: $\left(x+\frac{b}{2}\right)^2$. The completed perfect square trinomial form.

Flashcard 17: What value is added to $x^2+bx$ to complete the square?

Answer: $\left(\frac{b}{2}\right)^2$. Half the coefficient of $x$, then squared.

Flashcard 18: Identify the value added to complete the square in x^2-10x+__.

Answer: $25$. Half of $-10$ is $-5$, squared gives $25$.

Flashcard 19: What are the solutions of $(x-p)^2=q$ written explicitly?

Answer: $x=p\pm\sqrt{q}$. Explicit form after isolating $x$.

Flashcard 20: What does the symbol $\pm$ indicate when solving $(x-p)^2=q$?

Answer: Two cases: $+$ and $-$ square roots. Plus-minus accounts for both positive and negative square roots.

Flashcard 21: What is the goal form when completing the square for a quadratic in $x$?

Answer: $(x-p)^2 = q$. This standard form isolates the squared term and constant.

Flashcard 22: Rewrite $x^2-5x$ as a square plus a constant: x^2-5x=\left(x-__\right)^2-__.

Answer: $\left(x-\frac{5}{2}\right)^2-\frac{25}{4}$. Half of $-5$ is $-\frac{5}{2}$, then subtract $\left(\frac{5}{2}\right)^2$.

Flashcard 23: What square expression forms from $x^2+\frac{b}{a}x+\left(\frac{b}{2a}\right)^2$?

Answer: $\left(x+\frac{b}{2a}\right)^2$. The perfect square trinomial after completing.

Flashcard 24: After $x^2+\frac{b}{a}x=-\frac{c}{a}$, what is added to both sides to complete the square?

Answer: $\left(\frac{b}{2a}\right)^2$. Half the new coefficient of $x$, then squared.

Flashcard 25: What is the result after dividing $ax^2+bx+c=0$ by $a$?

Answer: $x^2+\frac{b}{a}x+\frac{c}{a}=0$. Standard form after dividing by the leading coefficient.

Flashcard 26: Solve by completing the square: $4x^2+4x-3=0$.

Answer: $x=\frac{1}{2}$ or $x=-\frac{3}{2}$. From $(x+\frac{1}{2})^2=1$, solve: $x=-\frac{1}{2}\pm 1$.

Flashcard 27: Transform $4x^2+4x-3=0$ into $(x-p)^2=q$ form.

Answer: $\left(x+\frac{1}{2}\right)^2=1$. First divide by $4$, then complete the square.

Flashcard 28: Solve by completing the square: $3x^2-12x+1=0$.

Answer: $x=2\pm\frac{\sqrt{33}}{3}$. From $(x-2)^2=\frac{11}{3}$, solve: $x=2\pm\sqrt{\frac{11}{3}}$.

Flashcard 29: Transform $3x^2-12x+1=0$ into $(x-p)^2=q$ form.

Answer: $\left(x-2\right)^2=\frac{11}{3}$. First divide by $3$, then complete the square.

Flashcard 30: Solve by completing the square: $2x^2+8x+3=0$.

Answer: $x=-2\pm\frac{\sqrt{10}}{2}$. From $(x+2)^2=\frac{5}{2}$, solve: $x=-2\pm\sqrt{\frac{5}{2}}$.