Algebra 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.

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: What is $p$ and $q$ if $x^2-6x+1=0$ is written as $\left(x-p\right)^2=q$?

Answer: $p=3,\ q=8$. From $x^2-6x+1=0$: $(x-3)^2=9-1=8$.

Flashcard 2: What operation isolates $x$ after obtaining $\left(x-p\right)^2=q$?

Answer: Take square roots: $x-p=\pm\sqrt{q}$. Taking square roots of both sides gives the plus-minus solutions.

Flashcard 3: What expression results after completing the square on $x^2+\frac{b}{a}x$?

Answer: $\left(x+\frac{b}{2a}\right)^2$. This is the perfect square formed after adding $\left(\frac{b}{2a}\right)^2$.

Flashcard 4: What are the solutions of $x^2+6x+5=0$?

Answer: $x=-1$ or $x=-5$. From $(x+3)^2=4$: $x+3=\pm 2$, so $x=-3\pm 2$.

Flashcard 5: What is the first step to complete the square for $ax^2+bx+c=0$ when $a\neq 1$?

Answer: Divide by $a$ so the $x^2$ coefficient is $1$. This makes the $x^2$ coefficient 1, simplifying the completing process.

Flashcard 6: What are the solutions of $x^2+4x-1=0$?

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

Flashcard 7: What are the solutions of $5x^2+20x+15=0$?

Answer: $x=-1$ or $x=-3$. From $(x+2)^2=1$: $x+2=\pm 1$, so $x=-2\pm 1$.

Flashcard 8: What property justifies dividing both sides by $a\neq 0$ in $ax^2+bx+c=0$?

Answer: Division Property of Equality. This allows dividing both sides by the same nonzero value.

Flashcard 9: What are the solutions of $x^2+6x+13=0$?

Answer: $x=-3\pm 2i$. From $(x+3)^2=-4$: $x=-3\pm\sqrt{-4}=-3\pm 2i$.

Flashcard 10: What is the completed-square form of $3x^2-12x+9=0$?

Answer: $\left(x-2\right)^2=1$. First divide by 3, then complete the square on $x^2-4x+3=0$.

Flashcard 11: What are the solutions of $2x^2-4x-1=0$?

Answer: $x=1\pm\frac{\sqrt{6}}{2}$. From $(x-1)^2=\frac{3}{2}$: $x=1\pm\sqrt{\frac{3}{2}}=1\pm\frac{\sqrt{6}}{2}$.

Flashcard 12: What are the solutions of $2x^2+8x+6=0$?

Answer: $x=-1$ or $x=-3$. From $(x+2)^2=1$: $x+2=\pm 1$, so $x=-2\pm 1$.

Flashcard 13: What equation results just before taking square roots when deriving the quadratic formula?

Answer: $\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}$. This form leads directly to the quadratic formula when square roots are taken.

Flashcard 14: What is the completed-square form of $3x^2-12x+9=0$?

Answer: $\left(x-2\right)^2=1$. First divide by 3, then complete the square on $x^2-4x+3=0$.

Flashcard 15: What is $p$ and $q$ if $x^2+10x+7=0$ is written as $\left(x-p\right)^2=q$?

Answer: $p=-5,\ q=18$. From $x^2+10x+7=0$: $(x+5)^2=25-7=18$.

Flashcard 16: What are the solutions of $\left(x-\frac{1}{2}\right)^2=\frac{25}{4}$?

Answer: $x=3$ or $x=-2$. From $x-\frac{1}{2}=\pm\frac{5}{2}$: $x=\frac{1}{2}\pm\frac{5}{2}$.

Flashcard 17: What is the completed-square form of $x^2+4x-1=0$?

Answer: $\left(x+2\right)^2=5$. Adding $\left(\frac{4}{2}\right)^2=4$ to both sides, then simplifying.

Flashcard 18: What property justifies adding the same number to both sides while completing the square?

Answer: Addition Property of Equality. This allows adding the same value to both sides of an equation.

Flashcard 19: What perfect-square identity rewrites $x^2+bx+\left(\frac{b}{2}\right)^2$?

Answer: $\left(x+\frac{b}{2}\right)^2$. This factors as a perfect square with half the linear coefficient.

Flashcard 20: What is the completed-square form of $5x^2+20x+15=0$?

Answer: $\left(x+2\right)^2=1$. First divide by 5: $x^2+4x+3=0$, then complete the square.

Flashcard 21: What are the solutions of $\left(x-3\right)^2=16$?

Answer: $x=7$ or $x=-1$. Solving $x-3=4$ and $x-3=-4$ gives these solutions.

Flashcard 22: What are the solutions of $x^2+6x+13=0$?

Answer: $x=-3\pm 2i$. From $(x+3)^2=-4$: $x=-3\pm\sqrt{-4}=-3\pm 2i$.

Flashcard 23: What are the solutions of $x^2-2x-8=0$?

Answer: $x=4$ or $x=-2$. From $(x-1)^2=9$: $x-1=\pm 3$, so $x=1\pm 3$.

Flashcard 24: What is the completed-square form of $4x^2+4x-3=0$?

Answer: $\left(x+\frac{1}{2}\right)^2=1$. First divide by 4, then complete the square on $x^2+x-\frac{3}{4}=0$.

Flashcard 25: What conclusion about real solutions follows from $\left(x+1\right)^2=-9$?

Answer: No real solutions. A perfect square cannot equal a negative real number.

Flashcard 26: What are the solutions of $4x^2+4x-3=0$?

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

Flashcard 27: What is the missing term to complete the square: $x^2-\frac{7}{3}x+\square=\left(x-\frac{7}{6}\right)^2$?

Answer: $\frac{49}{36}$. The missing term is $\left(\frac{-7/3}{2}\right)^2=\left(\frac{-7}{6}\right)^2=\frac{49}{36}$.

Flashcard 28: What are the solutions of $x^2-2x-8=0$?

Answer: $x=4$ or $x=-2$. From $(x-1)^2=9$: $x-1=\pm 3$, so $x=1\pm 3$.

Flashcard 29: What equation results just before taking square roots when deriving the quadratic formula?

Answer: $\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}$. This form leads directly to the quadratic formula when square roots are taken.

Flashcard 30: What are the solutions of $\left(x-3\right)^2=16$?

Answer: $x=7$ or $x=-1$. Solving $x-3=4$ and $x-3=-4$ gives these solutions.

Flashcard 31: What is the completed-square form of $4x^2+4x-3=0$?

Answer: $\left(x+\frac{1}{2}\right)^2=1$. First divide by 4, then complete the square on $x^2+x-\frac{3}{4}=0$.

Flashcard 32: What is the completed-square form of $x^2-10x+1=0$?

Answer: $\left(x-5\right)^2=24$. Adding $\left(\frac{-10}{2}\right)^2=25$ to both sides, then simplifying.

Flashcard 33: What perfect-square identity rewrites $x^2+bx+\left(\frac{b}{2}\right)^2$?

Answer: $\left(x+\frac{b}{2}\right)^2$. This factors as a perfect square with half the linear coefficient.

Flashcard 34: What value is added to $x^2+bx$ to complete the square?

Answer: $\left(\frac{b}{2}\right)^2$. Half the coefficient of $x$ squared makes the expression a perfect square trinomial.

Flashcard 35: What value is added to $x^2+bx$ to complete the square?

Answer: $\left(\frac{b}{2}\right)^2$. Half the coefficient of $x$ squared makes the expression a perfect square trinomial.

Flashcard 36: What is the completed-square form of $x^2-8x+7=0$?

Answer: $\left(x-4\right)^2=9$. Adding $\left(\frac{-8}{2}\right)^2=16$ to both sides, then simplifying.

Flashcard 37: What expression equals $\left(x+p\right)^2$ when expanded?

Answer: $x^2+2px+p^2$. This is the expanded form of the perfect square $(x+p)^2$.

Flashcard 38: What is the completed-square form of $x^2+6x+13=0$?

Answer: $\left(x+3\right)^2=-4$. Completing the square: $(x+3)^2=9-13=-4$.

Flashcard 39: What expression equals $\left(x-p\right)^2$ when expanded?

Answer: $x^2-2px+p^2$. This is the expanded form of the perfect square $(x-p)^2$.

Flashcard 40: What is the first step to complete the square for $ax^2+bx+c=0$ when $a\neq 1$?

Answer: Divide by $a$ so the $x^2$ coefficient is $1$. This makes the $x^2$ coefficient 1, simplifying the completing process.

Flashcard 41: What is the missing term to complete the square: $x^2+\frac{5}{2}x+\square=\left(x+\frac{5}{4}\right)^2$?

Answer: $\frac{25}{16}$. The missing term is $\left(\frac{5/2}{2}\right)^2=\left(\frac{5}{4}\right)^2=\frac{25}{16}$.

Flashcard 42: What is the value of $p$ in $\left(x-p\right)^2=q$ in terms of the linear coefficient $b$ of $x^2+bx+c$?

Answer: $p=-\frac{b}{2}$. The value of $p$ is negative half the linear coefficient.

Flashcard 43: What are the solutions of $x^2-8x+7=0$?

Answer: $x=1$ or $x=7$. From $(x-4)^2=9$: $x-4=\pm 3$, so $x=4\pm 3$.

Flashcard 44: What is the vertex form of $y=x^2+bx+c$ after completing the square?

Answer: $y=\left(x+\frac{b}{2}\right)^2+\left(c-\frac{b^2}{4}\right)$. This is vertex form with the vertex at $(-\frac{b}{2}, c-\frac{b^2}{4})$.

Flashcard 45: What is the missing term to complete the square: $x^2-12x+\square=\left(x-6\right)^2$?

Answer: $36$. The missing term is $\left(\frac{-12}{2}\right)^2=(-6)^2=36$.

Flashcard 46: What is the goal form when completing the square for a quadratic in $x$?

Answer: $(x-p)^2=q$. This is the target form with a perfect square equal to a constant.

Flashcard 47: What is the completed-square form of $5x^2+20x+15=0$?

Answer: $\left(x+2\right)^2=1$. First divide by 5: $x^2+4x+3=0$, then complete the square.

Flashcard 48: What is the completed-square form of $x^2-2x-8=0$?

Answer: $\left(x-1\right)^2=9$. Completing the square: $(x-1)^2=1+8=9$.

Flashcard 49: What are the solutions of $2x^2+8x+6=0$?

Answer: $x=-1$ or $x=-3$. From $(x+2)^2=1$: $x+2=\pm 1$, so $x=-2\pm 1$.

Flashcard 50: What is the correct square-root step for $\left(x-3\right)^2=16$?

Answer: $x-3=\pm 4$. Taking the square root of both sides: $\sqrt{16} = 4$.

Flashcard 51: What is the completed-square form of $2x^2-4x-1=0$?

Answer: $\left(x-1\right)^2=\frac{3}{2}$. First divide by 2: $x^2-2x-\frac{1}{2}=0$, then complete the square.

Flashcard 52: What is the completed-square form of $x^2+2x+10=0$?

Answer: $\left(x+1\right)^2=-9$. Completing the square gives a perfect square equal to a negative number.

Flashcard 53: What are the solutions of $5x^2+20x+15=0$?

Answer: $x=-1$ or $x=-3$. From $(x+2)^2=1$: $x+2=\pm 1$, so $x=-2\pm 1$.

Flashcard 54: What does a negative discriminant $b^2-4ac<0$ indicate about the solutions?

Answer: No real solutions (two complex solutions). Negative discriminant means taking the square root of a negative number.

Flashcard 55: What conclusion about real solutions follows from $\left(x+1\right)^2=-9$?

Answer: No real solutions. A perfect square cannot equal a negative real number.

Flashcard 56: What is the completed-square form of $2x^2+8x+6=0$?

Answer: $\left(x+2\right)^2=1$. First divide by 2, then complete the square on $x^2+4x+3=0$.

Flashcard 57: What are the solutions of $3x^2-12x+9=0$?

Answer: $x=1$ or $x=3$. From $(x-2)^2=1$: $x-2=\pm 1$, so $x=2\pm 1$.

Flashcard 58: What does a positive discriminant $b^2-4ac>0$ indicate about the solutions?

Answer: Two distinct real solutions. A positive discriminant means the square root is real with two values.

Flashcard 59: What does a zero discriminant $b^2-4ac=0$ indicate about the solutions?

Answer: One real solution (a double root). Zero discriminant means the square root is zero, giving one solution.

Flashcard 60: What property justifies dividing both sides by $a\neq 0$ in $ax^2+bx+c=0$?

Answer: Division Property of Equality. This allows dividing both sides by the same nonzero value.

Flashcard 61: What is the completed-square form of $2x^2+8x+6=0$?

Answer: $\left(x+2\right)^2=1$. First divide by 2, then complete the square on $x^2+4x+3=0$.

Flashcard 62: What is the completed-square form of $x^2-2x-8=0$?

Answer: $\left(x-1\right)^2=9$. Completing the square: $(x-1)^2=1+8=9$.

Flashcard 63: What is the vertex form of $y=x^2+bx+c$ after completing the square?

Answer: $y=\left(x+\frac{b}{2}\right)^2+\left(c-\frac{b^2}{4}\right)$. This is vertex form with the vertex at $(-\frac{b}{2}, c-\frac{b^2}{4})$.

Flashcard 64: What is the missing term to complete the square: $x^2-12x+\square=\left(x-6\right)^2$?

Answer: $36$. The missing term is $\left(\frac{-12}{2}\right)^2=(-6)^2=36$.

Flashcard 65: What is the completed-square form of $x^2+4x-1=0$?

Answer: $\left(x+2\right)^2=5$. Adding $\left(\frac{4}{2}\right)^2=4$ to both sides, then simplifying.

Flashcard 66: What is $p$ and $q$ if $x^2+10x+7=0$ is written as $\left(x-p\right)^2=q$?

Answer: $p=-5, q=18$. From $x^2+10x+7=0$: $(x+5)^2=25-7=18$.

Flashcard 67: What is the key added term after dividing by $a$ in $ax^2+bx+c=0$ to complete the square?

Answer: $\left(\frac{b}{2a}\right)^2$. After dividing by $a$, add half of the new linear coefficient squared.

Flashcard 68: What expression equals $\left(x+p\right)^2$ when expanded?

Answer: $x^2+2px+p^2$. This is the expanded form of the perfect square $(x+p)^2$.

Flashcard 69: What is the completed-square form of $x^2-10x+1=0$?

Answer: $\left(x-5\right)^2=24$. Adding $\left(\frac{-10}{2}\right)^2=25$ to both sides, then simplifying.

Flashcard 70: What is the quadratic formula for solutions of $ax^2+bx+c=0$?

Answer: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. This formula solves any quadratic equation and comes from completing the square.

Flashcard 71: What does a positive discriminant $b^2-4ac>0$ indicate about the solutions?

Answer: Two distinct real solutions. A positive discriminant means the square root is real with two values.

Flashcard 72: What is $p$ and $q$ if $x^2-6x+1=0$ is written as $\left(x-p\right)^2=q$?

Answer: $p=3,\ q=8$. From $x^2-6x+1=0$: $(x-3)^2=9-1=8$.

Flashcard 73: What is the correct square-root step for $\left(x-3\right)^2=16$?

Answer: $x-3=\pm 4$. Taking the square root of both sides: $\sqrt{16} = 4$.

Flashcard 74: What property justifies adding the same number to both sides while completing the square?

Answer: Addition Property of Equality. This allows adding the same value to both sides of an equation.

Flashcard 75: What are the solutions of $3x^2-12x+9=0$?

Answer: $x=1$ or $x=3$. From $(x-2)^2=1$: $x-2=\pm 1$, so $x=2\pm 1$.

Flashcard 76: What is the completed-square form of $x^2+6x+5=0$?

Answer: $\left(x+3\right)^2=4$. Adding $\left(\frac{6}{2}\right)^2=9$ to both sides, then moving constants.

Flashcard 77: What is the discriminant of $ax^2+bx+c=0$?

Answer: $b^2-4ac$. This expression determines the nature of the quadratic's solutions.

Flashcard 78: What expression equals $\left(x-p\right)^2$ when expanded?

Answer: $x^2-2px+p^2$. This is the expanded form of the perfect square $(x-p)^2$.

Flashcard 79: What is the completed-square form of $x^2+6x+13=0$?

Answer: $\left(x+3\right)^2=-4$. Completing the square: $(x+3)^2=9-13=-4$.

Flashcard 80: What does a negative discriminant $b^2-4ac<0$ indicate about the solutions?

Answer: No real solutions (two complex solutions). Negative discriminant means taking the square root of a negative number.

Flashcard 81: What are the solutions of $4x^2+4x-3=0$?

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

Flashcard 82: What operation isolates $x$ after obtaining $\left(x-p\right)^2=q$?

Answer: Take square roots: $x-p=\pm\sqrt{q}$. Taking square roots of both sides gives the plus-minus solutions.

Flashcard 83: What are the solutions of $x^2-10x+1=0$?

Answer: $x=5\pm 2\sqrt{6}$. From $(x-5)^2=24$: $x=5\pm\sqrt{24}=5\pm 2\sqrt{6}$.

Flashcard 84: What expression results after completing the square on $x^2+\frac{b}{a}x$?

Answer: $\left(x+\frac{b}{2a}\right)^2$. This is the perfect square formed after adding $\left(\frac{b}{2a}\right)^2$.

Flashcard 85: What are the solutions of $\left(x+2\right)^2=9$?

Answer: $x=1$ or $x=-5$. From $(x+2)^2=9$: $x+2=\pm 3$, so $x=-2\pm 3$.