This question tests your understanding of complex numbers and how to solve quadratic equations that have no real solutions but do have complex solutions involving the imaginary unit i. Complex solutions appear in conjugate pairs for quadratics with real coefficients: if 2 + 3i is a solution, then 2 - 3i is automatically also a solution (same real part, opposite imaginary part). This pairing is guaranteed by the ± in the quadratic formula and has important implications: the sum is real (4 + 6i + 4 - 6i = 8), and the product is real ((2+3i)(2-3i) = 4 + 9 = 13), which is why quadratics with complex roots can still have real coefficients! Solving x² - 6x + 10 = 0 using the quadratic formula: (1) Identify a = 1, b = -6, c = 10. (2) Calculate discriminant: b² - 4ac = (-6)² - 4(1)(10) = 36 - 40 = -4. (3) Since discriminant is negative, solutions are complex. (4) Apply formula: x = (6 ± √(-4))/2 = (6 ± 2i)/2. (5) Simplify: x = 6/2 ± 2i/2 = 3 ± i. Solutions: x = 3 + i and x = 3 - i. Choice A correctly solves to get x = 3 ± i with accurate calculation using i² = -1. Choice D forgets the i in the solution: when the discriminant is negative, we have √(negative) = i√(positive), so the i must appear in the answer. Without the i, these aren't complex numbers—they're just incorrect real numbers. The i is essential! Conjugate pair shortcut: if one solution is a + bi, immediately write down a - bi as the other without recalculating! For real-coefficient quadratics, complex solutions ALWAYS come in conjugate pairs. If the quadratic formula gives you (6 + 2i)/2 = 3 + i from the + version, the - version automatically gives 3 - i. Same real part, flip the imaginary sign. Done!

This question tests your understanding of complex numbers and how to solve quadratic equations that have no real solutions but do have complex solutions involving the imaginary unit i. When a quadratic equation has discriminant b² - 4ac < 0 (negative), the quadratic formula gives √(negative), which means complex solutions: x = (-b ± √(b² - 4ac))/(2a) = (-b ± i√|b² - 4ac|)/(2a). We separate the real part (-b/(2a)) from the imaginary part (±√|b² - 4ac|/(2a))i to write in a + bi form. The solutions always come in conjugate pairs: a + bi and a - bi. Solving x² + 8x + 20 = 0 using the quadratic formula: (1) Identify a = 1, b = 8, c = 20. (2) Calculate discriminant: b² - 4ac = (8)² - 4(1)(20) = 64 - 80 = -16. (3) Since discriminant is negative, solutions are complex. (4) Apply formula: x = (-8 ± √(-16))/2 = (-8 ± 4i)/2. (5) Simplify: x = -8/2 ± 4i/2 = -4 ± 2i. Solutions: x = -4 + 2i and x = -4 - 2i. Choice A correctly solves to get x = -4 ± 2i with accurate calculation using i² = -1. Choice B makes an error in the imaginary part: √(-16) = 4i, which when divided by 2 gives 2i, not 4i. When simplifying complex solutions, divide both the real and imaginary parts by the denominator: (-8 ± 4i)/2 = -8/2 ± 4i/2 = -4 ± 2i. Complex solution procedure: (1) Use quadratic formula or completing the square as usual, (2) When you get √(negative number), write it as i√(positive number): √(-16) = i√16 = 4i, √(-5) = i√5, (3) Simplify the entire expression to a + bi form by separating real parts from i-parts, (4) Write both solutions using ±: x = a ± bi means x = a + bi and x = a - bi. That's it! Complex solutions follow the same solving process, just with i appearing when discriminant is negative.

This question tests your understanding of complex numbers and how to solve quadratic equations that have no real solutions but do have complex solutions involving the imaginary unit i. When a quadratic equation has discriminant b² - 4ac < 0 (negative), the quadratic formula gives √(negative), which means complex solutions: x = (-b ± √(b² - 4ac))/(2a) = (-b ± i√|b² - 4ac|)/(2a). We separate the real part (-b/(2a)) from the imaginary part (±√|b² - 4ac|/(2a))i to write in a + bi form. The solutions always come in conjugate pairs: a + bi and a - bi. Solving x² + 10x + 34 = 0 using the quadratic formula: (1) Identify a = 1, b = 10, c = 34. (2) Calculate discriminant: b² - 4ac = (10)² - 4(1)(34) = 100 - 136 = -36. (3) Since discriminant is negative, solutions are complex. (4) Apply formula: x = (-10 ± √(-36))/2 = (-10 ± 6i)/2. (5) Simplify: x = -10/2 ± (6/2)i = -5 ± 3i. Solutions: x = -5 + 3i and x = -5 - 3i. Choice A correctly solves to get x = -5 ± 3i with accurate calculation using i² = -1. Choice D gives real solutions when the discriminant is negative. Check: b² - 4ac = 100 - 136 = -36. Negative discriminant means NO real solutions—only complex ones. When you get √(negative) in the formula, you can't ignore it or treat it as positive. You must include i! The discriminant is your early warning system: before solving, check b² - 4ac. Positive = two real solutions (no i needed). Zero = one real solution (perfect square). Negative = two complex solutions (i will appear). This tells you what to expect! If discriminant is -36 and you end up with no i in your answer, something went wrong.

This question tests your understanding of complex numbers and how to solve quadratic equations that have no real solutions but do have complex solutions involving the imaginary unit i. Complex solutions appear in conjugate pairs for quadratics with real coefficients: if 2 + 3i is a solution, then 2 - 3i is automatically also a solution (same real part, opposite imaginary part). This pairing is guaranteed by the ± in the quadratic formula and has important implications: the sum is real (4 + 6i + 4 - 6i = 8), and the product is real ((2+3i)(2-3i) = 4 + 9 = 13), which is why quadratics with complex roots can still have real coefficients! Solving x² - 4x + 5 = 0 using the quadratic formula: (1) Identify a = 1, b = -4, c = 5. (2) Calculate discriminant: (-4)² - 4(1)(5) = 16 - 20 = -4. (3) Since discriminant is negative, solutions are complex. (4) Apply formula: x = (4 ± √(-4))/2 = (4 ± i√4)/2. (5) Simplify: x = 4/2 ± (2i)/2 = 2 ± i. Solutions: x = 2 + i and x = 2 - i. Choice C correctly identifies the conjugate pair with accurate calculation using i² = -1. Choice A has the real and imaginary parts switched or calculated wrong: from the quadratic formula x = (-b ± i√|discriminant|)/(2a), the real part is -b/(2a) = 2, not what A has, and the imaginary part is √|discriminant|/(2a) = 1, not 2; keep real and imaginary separate in your calculation! Simplifying with i: remember that i² = -1 is the key rule. When you expand (a + bi)², you get a² + 2abi + (bi)² = a² + 2abi + b²i² = a² + 2abi - b² (since i² = -1). Combine reals and imaginaries separately: (a² - b²) + (2ab)i. This is how verification works—expand, use i² = -1, group real and imaginary parts. Don't fear complex solutions: they're not 'worse' than real solutions, just different! In Algebra 1, you're learning that every quadratic has solutions—sometimes real (discriminant ≥ 0), sometimes complex (discriminant < 0). The complex solutions are just as valid mathematically. They don't represent x-intercepts on a standard graph, but they're still solutions to the equation!

This question tests your understanding of complex numbers and how to solve quadratic equations that have no real solutions but do have complex solutions involving the imaginary unit i. Complex solutions appear in conjugate pairs for quadratics with real coefficients: if 2 + 3i is a solution, then 2 - 3i is automatically also a solution (same real part, opposite imaginary part). This pairing is guaranteed by the ± in the quadratic formula and has important implications: the sum is real (4 + 6i + 4 - 6i = 8), and the product is real ((2+3i)(2-3i) = 4 + 9 = 13), which is why quadratics with complex roots can still have real coefficients! Solving x² - 2x + 10 = 0 using the quadratic formula: (1) Identify a = 1, b = -2, c = 10. (2) Calculate discriminant: (-2)² - 4(1)(10) = 4 - 40 = -36. (3) Since discriminant is negative, solutions are complex. (4) Apply formula: x = (2 ± √(-36))/2 = (2 ± i√36)/2. (5) Simplify: x = 2/2 ± (6i)/2 = 1 ± 3i. Solutions: x = 1 + 3i and x = 1 - 3i. Choice A correctly identifies the conjugate pair with accurate calculation using i² = -1. Choice B doesn't simplify the radical: √36 under the i should be simplified to 6i/2=3i, but B has 1 ± i which is too small; even with complex numbers, we still simplify radicals! √(-36) = i√36 = i·6 = 6i, not i√36 in final form. Take that extra step to simplify. The discriminant is your early warning system: before solving, check b² - 4ac. Positive = two real solutions (no i needed). Zero = one real solution (perfect square). Negative = two complex solutions (i will appear). This tells you what to expect! If discriminant is -20 and you end up with no i in your answer, something went wrong. Conjugate pair shortcut: if one solution is a + bi, immediately write down a - bi as the other without recalculating! For real-coefficient quadratics, complex solutions ALWAYS come in conjugate pairs. If the quadratic formula gives you (-2 + 4i)/2 = -1 + 2i from the + version, the - version automatically gives -1 - 2i. Same real part, flip the imaginary sign. Done!

This question tests your understanding of complex numbers and how to solve quadratic equations that have no real solutions but do have complex solutions involving the imaginary unit i. The imaginary unit i is defined as i = √(-1), which means i² = -1: this allows us to take square roots of negative numbers. When we have √(-16), we can write it as i√16 = 4i. Complex numbers have the form a + bi where a is the real part and b is the imaginary part (like 3 + 2i or -1 - 4i). These aren't 'imaginary' in the sense of not existing—they're a real extension of the number system that makes every quadratic equation solvable! Solving x² + 2x + 5 = 0 using the quadratic formula: (1) Identify a = 1, b = 2, c = 5. (2) Calculate discriminant: b² - 4ac = (2)² - 4(1)(5) = 4 - 20 = -16. (3) Since discriminant is negative, solutions are complex. (4) Apply formula: x = (-2 ± √(-16))/2 = (-2 ± 4i)/2. (5) Simplify: x = -2/2 ± 4i/2 = -1 ± 2i. Solutions: x = -1 + 2i and x = -1 - 2i. Choice A correctly solves to get x = -1 ± 2i with accurate calculation using i² = -1. Choice B has the wrong real part: from the quadratic formula x = (-b ± i√|discriminant|)/(2a), the real part is -b/(2a) = -2/2 = -1, not 1, and the imaginary part is √16/2 = 4/2 = 2. Keep real and imaginary separate in your calculation! Complex solution procedure: (1) Use quadratic formula or completing the square as usual, (2) When you get √(negative number), write it as i√(positive number): √(-16) = i√16 = 4i, √(-5) = i√5, (3) Simplify the entire expression to a + bi form by separating real parts from i-parts, (4) Write both solutions using ±: x = a ± bi means x = a + bi and x = a - bi. That's it! Complex solutions follow the same solving process, just with i appearing when discriminant is negative.

This question tests your understanding of complex numbers and how to solve quadratic equations that have no real solutions but do have complex solutions involving the imaginary unit i. The imaginary unit i is defined as i = √(-1), which means i² = -1: this allows us to take square roots of negative numbers. When we have √(-16), we can write it as i√16 = 4i. Complex numbers have the form a + bi where a is the real part and b is the imaginary part (like 3 + 2i or -1 - 4i). These aren't 'imaginary' in the sense of not existing—they're a real extension of the number system that makes every quadratic equation solvable! Solving x² + 2x + 2 = 0 using the quadratic formula: (1) Identify a = 1, b = 2, c = 2. (2) Calculate discriminant: 2² - 4(1)(2) = 4 - 8 = -4. (3) Since discriminant is negative, solutions are complex. (4) Apply formula: x = (-2 ± √(-4))/2 = (-2 ± i√4)/2. (5) Simplify: x = -2/2 ± (2i)/2 = -1 ± i. Solutions: x = -1 + i and x = -1 - i. Choice A correctly solves to get x = -1 ± i with accurate calculation using i² = -1. Choice C has the real and imaginary parts switched or calculated wrong: the imaginary part is 1, not 2; from the quadratic formula x = (-b ± i√|discriminant|)/(2a), the real part is -b/(2a) = -1, and the imaginary part is √|discriminant|/(2a) = 1; keep real and imaginary separate in your calculation! The discriminant is your early warning system: before solving, check b² - 4ac. Positive = two real solutions (no i needed). Zero = one real solution (perfect square). Negative = two complex solutions (i will appear). This tells you what to expect! If discriminant is -20 and you end up with no i in your answer, something went wrong. Don't fear complex solutions: they're not 'worse' than real solutions, just different! In Algebra 1, you're learning that every quadratic has solutions—sometimes real (discriminant ≥ 0), sometimes complex (discriminant < 0). The complex solutions are just as valid mathematically. They don't represent x-intercepts on a standard graph, but they're still solutions to the equation!

This question tests your understanding of complex numbers and how to solve quadratic equations that have no real solutions but do have complex solutions involving the imaginary unit i. When a quadratic equation has discriminant b² - 4ac < 0 (negative), the quadratic formula gives √(negative), which means complex solutions: x = (-b ± √(b² - 4ac))/(2a) = (-b ± i√|b² - 4ac|)/(2a). We separate the real part (-b/(2a)) from the imaginary part (±√|b² - 4ac|/(2a))i to write in a + bi form. The solutions always come in conjugate pairs: a + bi and a - bi. Solving x² - 4x + 5 = 0 by completing the square: (1) Move constant: x² - 4x = -5. (2) Add (b/2)²: x² - 4x + 4 = -5 + 4 = -1. (3) Factor: (x - 2)² = -1. (4) Take square roots: x - 2 = ±√(-1) = ±i. (5) Solve: x = 2 ± i. The negative value after completing the square signals complex solutions! Choice B correctly identifies x = 2 ± i with accurate calculation using i² = -1. Choice D gives real solutions when the discriminant is negative. Check: b² - 4ac = (-4)² - 4(1)(5) = 16 - 20 = -4. Negative discriminant means NO real solutions—only complex ones. When you get √(negative) in the formula, you can't ignore it or treat it as positive. You must include i! The discriminant is your early warning system: before solving, check b² - 4ac. Positive = two real solutions (no i needed). Zero = one real solution (perfect square). Negative = two complex solutions (i will appear). This tells you what to expect! If discriminant is -4 and you end up with no i in your answer, something went wrong.

This question tests your understanding of complex numbers and how to solve quadratic equations that have no real solutions but do have complex solutions involving the imaginary unit i. The imaginary unit i is defined as i = $\sqrt{-1}$, which means $i^2 = -1$: this allows us to take square roots of negative numbers. When we have $\sqrt{-16}$, we can write it as $i \sqrt{16} = 4i$. Complex numbers have the form $a + bi$ where a is the real part and b is the imaginary part (like $3 + 2i$ or $-1 - 4i$). These aren't 'imaginary' in the sense of not existing—they're a real extension of the number system that makes every quadratic equation solvable! Solving $x^2 + 6x + 25 = 0$ using the quadratic formula: (1) Identify a = 1, b = 6, c = 25. (2) Calculate discriminant: $b^2 - 4ac = 6^2 - 4(1)(25) = 36 - 100 = -64$. (3) Since discriminant is negative, solutions are complex. (4) Apply formula: $x = (-6 \pm \sqrt{-64})/2 = (-6 \pm 8i)/2$. (5) Simplify: $x = -6/2 \pm 8i/2 = -3 \pm 4i$. Solutions: $x = -3 + 4i$ and $x = -3 - 4i$. Choice A correctly solves to get $x = -3 \pm 4i$ with accurate calculation using $i^2 = -1$. Choice D forgets the i in the solution: when the discriminant is negative, we have $\sqrt{\text{negative}} = i \sqrt{\text{positive}}$, so the i must appear in the answer. Without the i, these aren't complex numbers—they're just incorrect real numbers. The i is essential! Complex solution procedure: (1) Use quadratic formula or completing the square as usual, (2) When you get $\sqrt{\text{negative number}}$, write it as $i \sqrt{\text{positive number}}$: $\sqrt{-16} = i \sqrt{16} = 4i$, $\sqrt{-5} = i \sqrt{5}$, (3) Simplify the entire expression to $a + bi$ form by separating real parts from i-parts, (4) Write both solutions using $\pm$: $x = a \pm bi$ means $x = a + bi$ and $x = a - bi$. That's it! Complex solutions follow the same solving process, just with i appearing when discriminant is negative.

This question tests your understanding of complex numbers and how to solve quadratic equations that have no real solutions but do have complex solutions involving the imaginary unit i. Complex solutions appear in conjugate pairs for quadratics with real coefficients: if 2 + 3i is a solution, then 2 - 3i is automatically also a solution (same real part, opposite imaginary part). This pairing is guaranteed by the ± in the quadratic formula and has important implications: the sum is real (4 + 6i + 4 - 6i = 8), and the product is real ((2+3i)(2-3i) = 4 + 9 = 13), which is why quadratics with complex roots can still have real coefficients! Solving x² - 4x + 5 = 0 using the quadratic formula: (1) Identify a = 1, b = -4, c = 5. (2) Calculate discriminant: b² - 4ac = (-4)² - 4(1)(5) = 16 - 20 = -4. (3) Since discriminant is negative, solutions are complex. (4) Apply formula: x = (4 ± √(-4))/2 = (4 ± i√4)/2. (5) Simplify: x = 4/2 ± (√4/2)i = 2 ± 1i = 2 ± i. Solutions: x = 2 + i and x = 2 - i. Choice A correctly expresses in proper a + bi form x = 2 ± i with accurate calculation using i² = -1. Choice D forgets the i in the solution: when the discriminant is negative, we have √(negative) = i√(positive), so the i must appear in the answer. Without the i, these aren't complex numbers—they're just incorrect real numbers. The i is essential! Conjugate pair shortcut: if one solution is a + bi, immediately write down a - bi as the other without recalculating! For real-coefficient quadratics, complex solutions ALWAYS come in conjugate pairs. If the quadratic formula gives you (4 + 2i)/2 = 2 + i from the + version, the - version automatically gives 2 - i. Same real part, flip the imaginary sign. Done!