Algebra 2 Flashcards: Solve Quadratics By Multiple Methods

What this deck covers

This deck focuses on Solve Quadratics By Multiple Methods, 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: Factor and solve $x^2+7x+12=0$.

Answer: $x=-3$ or $x=-4$. Factor: $(x+3)(x+4)=0$, then use zero-product property.

Flashcard 2: Solve using the quadratic formula: $x^2+4x+1=0$.

Answer: $x=-2\pm\sqrt{3}$. Use $a=1$, $b=4$, $c=1$ in the quadratic formula.

Flashcard 3: Solve using the quadratic formula: $x^2+2x+5=0$.

Answer: $x=-1\pm 2i$. Negative discriminant gives complex solutions: $\sqrt{-16}=4i$.

Flashcard 4: Solve by taking square roots: $(x+2)^2=7$.

Answer: $x=-2\pm\sqrt{7}$. Square root both sides: $x+2=\pm\sqrt{7}$.

Flashcard 5: What is the zero-product property used after factoring $AB=0$?

Answer: $AB=0\Rightarrow A=0$ or $B=0$. If a product equals zero, at least one factor must be zero.

Flashcard 6: What does $\Delta<0$ guarantee about solutions of $ax^2+bx+c=0$?

Answer: Two complex conjugate solutions. Negative discriminant means the parabola doesn't cross the x-axis.

Flashcard 7: Solve using the quadratic formula: $2x^2+4x+5=0$.

Answer: $x=-1\pm\frac{\sqrt{6}}{2}i$. Use $a=2$, $b=4$, $c=5$; $\sqrt{-24}=2\sqrt{6}i$.

Flashcard 8: Solve by factoring: $3x^2-12=0$.

Answer: $x=\pm 2$. Factor: $3(x^2-4)=3(x-2)(x+2)=0$.

Flashcard 9: State the square root property used to solve $(x-h)^2=k$.

Answer: $(x-h)^2=k\Rightarrow x-h=\pm\sqrt{k}$. Take the square root of both sides to isolate $x-h$.

Flashcard 10: What is $\Delta$ for $x^2+2x+5=0$?

Answer: $\Delta=-16$. $\Delta=(2)^2-4(1)(5)=4-20=-16$.

Flashcard 11: Solve using the quadratic formula: $x^2+5x+6=0$.

Answer: $x=-2$ or $x=-3$. Use $a=1$, $b=5$, $c=6$ in the quadratic formula.

Flashcard 12: Solve by completing the square: $x^2-4x-1=0$.

Answer: $x=2\pm\sqrt{5}$. Complete the square: $(x-2)^2=5$, so $x-2=\pm\sqrt{5}$.

Flashcard 13: What are the solutions of $(x+5)^2=9$?

Answer: $x=-2$ or $x=-8$. Square root both sides: $x+5=\pm 3$, so $x=-5\pm 3$.

Flashcard 14: Solve by completing the square: $x^2+6x+1=0$.

Answer: $x=-3\pm 2\sqrt{2}$. Complete the square: $(x+3)^2=8$, so $x+3=\pm 2\sqrt{2}$.

Flashcard 15: State the quadratic formula for solutions of $ax^2+bx+c=0$.

Answer: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Standard form for finding roots of any quadratic equation.

Flashcard 16: What is the simplified form of $\sqrt{-12}$ in terms of $i$?

Answer: $2\sqrt{3}i$. Simplify: $\sqrt{-12}=\sqrt{12}i=2\sqrt{3}i$.

Flashcard 17: Solve by factoring: $x^2+10x+25=0$.

Answer: $x=-5$. Perfect square trinomial: $(x+5)^2=0$, so $x=-5$.

Flashcard 18: Solve using the quadratic formula: $3x^2-12x+12=0$.

Answer: $x=2$. Use $a=3$, $b=-12$, $c=12$; discriminant equals 0.

Flashcard 19: Factor and solve $x^2-5x=0$.

Answer: $x=0$ or $x=5$. Factor out $x$: $x(x-5)=0$, then use zero-product property.

Flashcard 20: What is the simplified form of $\sqrt{-1}$?

Answer: $i$. The imaginary unit where $i^2=-1$.

Flashcard 21: Factor and solve $x^2-9=0$.

Answer: $x=\pm 3$. Difference of squares: $(x-3)(x+3)=0$.

Flashcard 22: What are the solutions of $x^2=49$?

Answer: $x=\pm 7$. Take the square root of both sides: $\sqrt{49}=7$.

Flashcard 23: Solve by taking square roots: $(2x-1)^2=25$.

Answer: $x=3$ or $x=-2$. Square root both sides: $2x-1=\pm 5$, solve for $x$.

Flashcard 24: What does $\Delta>0$ guarantee about solutions of $ax^2+bx+c=0$?

Answer: Two distinct real solutions. Positive discriminant means the parabola crosses the x-axis twice.

Flashcard 25: What is $\Delta$ for $2x^2-3x-2=0$?

Answer: $\Delta=25$. $\Delta=(-3)^2-4(2)(-2)=9+16=25$.

Flashcard 26: What does $\Delta=0$ guarantee about solutions of $ax^2+bx+c=0$?

Answer: One real double solution. Zero discriminant means the parabola touches the x-axis at one point.

Flashcard 27: Solve by completing the square: $x^2+2x-8=0$.

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

Flashcard 28: Factor and solve $x^2-6x+8=0$.

Answer: $x=2$ or $x=4$. Factor: $(x-2)(x-4)=0$, then use zero-product property.

Flashcard 29: Identify the best method for $x^2-16=0$ from factoring or quadratic formula.

Answer: Factoring. Difference of squares factors easily as $(x-4)(x+4)=0$.

Flashcard 30: What is the simplified form of $\sqrt{-49}$ in terms of $i$?

Answer: $7i$. Use $\sqrt{-49}=\sqrt{49}\cdot\sqrt{-1}=7i$.

Flashcard 31: Identify $a$, $b$, and $c$ for $-3x^2+2x-7=0$.

Answer: $a=-3$, $b=2$, $c=-7$. Coefficients from standard form $ax^2+bx+c=0$.

Flashcard 32: Solve by factoring: $x^2-2x-15=0$.

Answer: $x=5$ or $x=-3$. Factor: $(x-5)(x+3)=0$, then use zero-product property.

Flashcard 33: What are the solutions of $(x-4)^2=0$?

Answer: $x=4$. Only one solution since the square root of 0 is 0.

Flashcard 34: What is the discriminant for $ax^2+bx+c=0$?

Answer: $\Delta=b^2-4ac$. The discriminant determines the nature of quadratic solutions.

Flashcard 35: What is $\Delta$ for $x^2-6x+9=0$?

Answer: $\Delta=0$. $\Delta=(-6)^2-4(1)(9)=36-36=0$.

Flashcard 36: Solve by inspection: $x^2=\frac{1}{4}$.

Answer: $x=\pm\frac{1}{2}$. Take the square root: $\sqrt{\frac{1}{4}}=\frac{1}{2}$.

Flashcard 37: Rewrite $x^2-12x+36$ as a perfect square trinomial.

Answer: $(x-6)^2$. Perfect square trinomial with $a=x$ and $b=-6$.

Flashcard 38: How do you write solutions when the quadratic formula yields $\sqrt{-k}$ for $k>0$?

Answer: Use $\sqrt{-k}=i\sqrt{k}$ to write $a\pm bi$. Convert negative square roots to imaginary form using $i$.

Flashcard 39: What number completes the square for $x^2-6x+\square$?

Answer: $9$. Half of -6 is -3, and $(-3)^2=9$.

Flashcard 40: Identify the best method for $(x-9)^2=12$ from square roots or quadratic formula.

Answer: Taking square roots. Already in perfect square form, ready for square root method.

Flashcard 41: What are the solutions of $(x-3)^2=16$?

Answer: $x=7$ or $x=-1$. Square root both sides: $x-3=\pm 4$, so $x=3\pm 4$.

Flashcard 42: Solve by factoring: $x^2-4x+3=0$.

Answer: $x=1$ or $x=3$. Factor: $(x-1)(x-3)=0$, then use zero-product property.

Flashcard 43: Rewrite $x^2+8x+16$ as a perfect square trinomial.

Answer: $(x+4)^2$. Perfect square trinomial with $a=x$ and $b=4$.

Flashcard 44: Factor and solve $x^2+x-6=0$.

Answer: $x=2$ or $x=-3$. Factor: $(x+3)(x-2)=0$, then use zero-product property.

Flashcard 45: Solve using the quadratic formula: $2x^2+3x-2=0$.

Answer: $x=\frac{1}{2}$ or $x=-2$. Use $a=2$, $b=3$, $c=-2$ in the quadratic formula.

Flashcard 46: What are the solutions of $x^2+8x+16=0$?

Answer: $x=-4$. Perfect square trinomial: $(x+4)^2=0$, so $x=-4$.

Flashcard 47: Solve using the quadratic formula: $x^2-2x+5=0$.

Answer: $x=1\pm 2i$. Negative discriminant gives complex solutions: $\sqrt{-16}=4i$.

Flashcard 48: Solve by factoring: $2x^2-8x=0$.

Answer: $x=0$ or $x=4$. Factor out $2x$: $2x(x-4)=0$, then use zero-product property.

Flashcard 49: What is the simplified form of $\sqrt{-18}$ in terms of $i$?

Answer: $3\sqrt{2}i$. Simplify: $\sqrt{-18}=\sqrt{18}i=3\sqrt{2}i$.

Flashcard 50: What number completes the square for $x^2+10x+\square$?

Answer: $25$. Half of 10 is 5, and $5^2=25$.

Flashcard 51: What are the solutions of $2(x-1)^2=18$?

Answer: $x=4$ or $x=-2$. Divide by 2 first: $(x-1)^2=9$, then $x-1=\pm 3$.

Flashcard 52: State the first step to complete the square for $x^2+bx$.

Answer: Add $\left(\frac{b}{2}\right)^2$. Take half the coefficient of $x$, then square it.