This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. For solving x squared + 4x + 8 = 0: identify a = 1, b = 4, c = 8. Calculate discriminant: b squared - 4ac = 16 - 32 = -16 (negative, so complex solutions!). Apply formula: x = (-4 plus or minus square root of -16) divided by 2 = (-4 plus or minus 4i) divided by 2 = -2 plus or minus 2i. The negative discriminant signals that i will appear in the answer. Choice A correctly identifies x = -2 plus or minus 2i as the complex solutions in a plus or minus bi form. Choice B incorrectly has x = 2 plus or minus 2i, forgetting that -b/2a = -4/2 = -2, not 2. When b = 4, we get -b = -4 in the numerator, and dividing by 2a = 2 gives -2 for the real part. Complex solution recognition: for x squared + 4x + 8 = 0, discriminant = 16 - 32 = -16 (negative!), so complex solutions: x = (-4 plus or minus square root of -16) divided by 2 = (-4 plus or minus 4i) divided by 2 = -2 plus or minus 2i. The pattern is clear: real part = -b/(2a) = -4/2 = -2, imaginary part = plus or minus square root of |discriminant|/(2a) = plus or minus 4/2 = plus or minus 2.

This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For (x-3) squared = 5, take square roots on both sides: x-3 = ± sqrt(5), so x = 3 ± sqrt(5)—remember the ± for both solutions. This method is perfect here since it's already in squared form. Choice A correctly applies the square root method, including both positive and negative roots. Choice D fails by omitting the positive root, giving only one solution—quadratics usually have two solutions, so always include ±! Remember, when taking square roots, if the right side is positive, you get two real solutions; if zero, one; if negative, complex. Practice by rewriting equations into squared form when possible for efficiency.

This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. For solving x squared - 6x + 10 = 0 with negative discriminant: identify a = 1, b = -6, c = 10. Calculate discriminant: b squared - 4ac = 36 - 40 = -4 (negative, so complex solutions!). Apply formula: x = (6 plus or minus square root of -4) divided by 2 = (6 plus or minus 2i) divided by 2 = 3 plus or minus i. The negative discriminant signals that i will appear in the answer. Choice A correctly identifies x = 3 plus or minus i as the complex solutions in a plus or minus bi form. Choice B incorrectly has x = -3 plus or minus i, which would come from using b = 6 instead of b = -6 in the formula. When b = -6, we get -b = -(-6) = 6 in the numerator, giving x = (6 plus or minus 2i) divided by 2 = 3 plus or minus i. Complex solution recognition: before solving, check the discriminant b squared - 4ac. Negative means two complex solutions (use formula or completing square, write as a plus or minus bi). For x squared - 6x + 10 = 0: discriminant = 36 - 40 = -4 (negative!), so complex solutions: x = (6 plus or minus square root of -4) divided by 2 = (6 plus or minus 2i) divided by 2 = 3 plus or minus i.

This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's $x^2 = \text{number}$, inspect or take square roots ($x = \pm \sqrt{\text{number}}$), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like $x^2 + 5x + 6$), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For $3x^2 + 6x + 5 = 0$: discriminant $36 - 60 = -24$ (negative, complex), formula $x = \frac{-6 \pm \sqrt{-24}}{6} = \frac{-6 \pm 2 \sqrt{6} , i}{6} = -1 \pm \frac{\sqrt{6}}{3} , i$. Choice B correctly computes and simplifies the complex form. Choice A omits i—check discriminant sign! Recognition: negative means $a \pm bi$. Impressive handling of complexes— you're advancing!

This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's $x^2 = \text{number}$, inspect or take square roots ($x = \pm \sqrt{\text{number}}$), (2) if it's $(\text{expression})^2 = \text{number}$, take square roots, (3) if it factors easily (like $x^2 + 5x + 6$), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For $2x^2 - 3x - 5 = 0$ using quadratic formula: $a=2$, $b=-3$, $c=-5$, discriminant $9 + 40=49$, $x=\frac{3 \pm 7}{4}$ (since $-b=3$). Choice A correctly plugs in with $-b$ positive and denominator 4. Choice B uses $+b$ instead of $-b$—double-check the formula sign! Transferable strategy: write $a$, $b$, $c$ clearly before plugging in. Great precision—you've got this!

This question targets completing the square, a method that rewrites quadratics as $(x + h)^2 = k$, then takes square roots—useful for deriving the formula or when factoring fails. For $x^2 + 6x + 1 = 0$, move 1: $x^2 + 6x = -1$, add $(\frac{6}{2})^2 = 9$: $(x+3)^2 = 8$, $x+3 = \pm \sqrt{8} = \pm 2\sqrt{2}$, $x = -3 \pm 2\sqrt{2}$—perfect! This method shines for monic quadratics without even coefficients. Choice C correctly simplifies $\sqrt{8}$ to $2\sqrt{2}$. Choice A forgets to simplify the radical—always reduce for clean answers. Practice by halving b and squaring to complete efficiently. You've got the hang of it—keep building!

This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For x squared + 4x + 8 = 0, discriminant = 16 - 32 = -16 (negative), so complex: using quadratic formula, x = [-4 ± sqrt(-16)] / 2 = [-4 ± 4i] / 2 = -2 ± 2i. Completing the square also works: x squared + 4x = -8, add 4: (x+2) squared = -4, x+2 = ± 2i, x = -2 ± 2i. Choice A correctly expresses the complex solutions in a ± bi form after proper calculation. Choice C is tempting but doubles the imaginary part incorrectly—remember to divide the entire numerator by 2a. Complex solution tip: factor out the square root of -1 as i, and simplify fractions. Always check discriminant first to predict solution type.

This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's $x^2 = \text{number}$, inspect or take square roots ($x = \pm \sqrt{\text{number}}$), (2) if it's $(\text{expression})^2 = \text{number}$, take square roots, (3) if it factors easily (like $x^2 + 5x + 6$), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For $(2x+1)^2 = 36$ by taking square roots: $2x+1 = \pm 6$, so $2x=5$ or $2x=-7$, $x=\frac{5}{2}$ or $x=-\frac{7}{2}$. Choice B correctly solves both branches. Choice A ignores the negative root—always consider both plus and minus! Strategy: isolate the squared term, then apply square roots with $\pm$. You're nailing efficiency—keep it up!

This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. For solving 2x squared + 3x - 7 = 0 by quadratic formula: identify a = 2, b = 3, c = -7. Calculate discriminant: b squared - 4ac = 9 - 4(2)(-7) = 9 + 56 = 65 (positive, so two real solutions). Apply formula: x = (-3 plus or minus square root of 65) divided by (2 times 2) = (-3 plus or minus square root of 65) divided by 4. Choice A correctly applies the quadratic formula with x = (-3 plus or minus square root of 65) divided by 4. Choice D incorrectly has denominator 2 instead of 2a = 4—remember the denominator is 2a, not just 2! When a = 2, the denominator becomes 2(2) = 4. This is a common error: forgetting to multiply by the coefficient 'a' in the denominator. Always use 2a in the denominator, where a is the coefficient of x squared. For 2x squared + 3x - 7 = 0, we have a = 2, so denominator = 2(2) = 4, giving x = (-3 plus or minus square root of 65) divided by 4.

This question asks for the most efficient method to solve a quadratic in $(expression)^2$ = number form, highlighting method selection based on equation structure. Taking square roots is quickest: for $(2x-1)^2$=25, 2x-1=±5, then 2x=6 or -4, x=3 or -2—efficient and direct! Expanding to standard form would work but adds unnecessary steps. Choice B correctly picks taking square roots for this form. Choice A, factoring, tempts if you expand first, but it's less efficient—stick to the given form. Use the decision tree: if squared equals constant, square root first. You're sharpening your skills—great job choosing wisely!