This question tests your ability to use completing the square to solve quadratic equations by transforming them into the form (x - p)² = q, which makes finding solutions straightforward by taking square roots. The quadratic formula x = (-b ± √(b² - 4ac))/(2a) actually comes FROM completing the square on the general equation ax² + bx + c = 0! If you complete the square on this general form (treating a, b, c as constants), you derive the quadratic formula. So completing the square isn't just one method among many—it's the fundamental method that gives us the formula itself! Starting with ax² + bx + c = 0: (1) Divide by a: x² + (b/a)x + (c/a) = 0. (2) Move constant: x² + (b/a)x = -c/a. (3) Complete square, half of b/a is b/(2a), squared is b²/(4a²): x² + (b/a)x + b²/(4a²) = -c/a + b²/(4a²). (4) Left side: (x + b/(2a))². Right side: (b² - 4ac)/(4a²). (5) So (x + b/(2a))² = (b² - 4ac)/(4a²). (6) Square root: x + b/(2a) = ±√(b² - 4ac)/(2a). (7) Solve: x = -b/(2a) ± √(b² - 4ac)/(2a) = (-b ± √(b² - 4ac))/(2a). That's the quadratic formula! Choice C correctly derives the formula x = (-b ± √(b² - 4ac))/(2a) with accurate algebra and proper signs throughout the derivation. Choice A has the wrong sign under the radical: when we move c/a to the right and add b²/(4a²), we get (b² - 4ac)/(4a²), not (b² + 4ac)/(4a²). The -4ac term comes from -c/a = -4ac/(4a²) when we find a common denominator. Sign errors in derivations are common but critical! Why this works: when you add (b/2)² to x² + bx, you're creating the expansion of (x + b/2)²: remember (a + b)² = a² + 2ab + b², so (x + b/2)² = x² + 2(x)(b/2) + (b/2)² = x² + bx + (b/2)². That's why adding (b/2)² makes a perfect square! It's not random—it completes the perfect square trinomial pattern.

This question tests your ability to use completing the square to solve quadratic equations by transforming them into the form (x - p)² = q, which makes finding solutions straightforward by taking square roots. Completing the square is a method that transforms any quadratic equation ax² + bx + c = 0 into an equivalent equation (x - p)² = q that has the same solutions but is much easier to solve: once in this form, we just take the square root of both sides (remembering ±!), giving x - p = ±√q, then solve for x = p ± √q. This method works for ALL quadratics, even ones that don't factor nicely! Solving x² + 8x + 7 = 0 by completing the square: (1) Move constant to right: x² + 8x = -7. (2) Take half of 8 to get 4, square it to get 16, add to both sides: x² + 8x + 16 = -7 + 16. (3) Left side is perfect square (x + 4)², right side simplifies to 9: (x + 4)² = 9. (4) Take square roots: x + 4 = ±√9. (5) Solve: x = -4 ± 3. Done! Choice B correctly completes the square to get (x + 4)² = 9 and solves to get x = -4 ± 3 with accurate arithmetic and proper form. Choice D forgets the ± when taking square roots: from (x + 4)² = 9, we get x + 4 = ±3 (both positive and negative square root), giving TWO solutions x = -4 + 3 and x = -4 - 3. Missing the ± means missing a solution! This is one of the most common mistakes in completing the square. The complete completing-the-square procedure: (1) If a ≠ 1, divide everything by a first to get x² coefficient = 1, (2) Move the constant term to the right side, (3) Take half the x-coefficient, square it: (b/2)², (4) Add this to BOTH sides, (5) Factor left side as (x + b/2)² (it's a perfect square now!), (6) Simplify right side, (7) Take ± square root of both sides, (8) Solve for x. Follow these 8 steps systematically and you'll get the right answer every time!

This question tests your ability to use completing the square to solve quadratic equations by transforming them into the form (x - p)² = q, which makes finding solutions straightforward by taking square roots. The form (x - p)² = q tells us immediately: if q > 0, there are two real solutions (x = p ± √q); if q = 0, there's one solution (x = p); if q < 0, there are no real solutions (we'd need to take square root of negative). So completing the square not only solves the equation, it also reveals the nature of the solutions before we calculate them! Solving x² + 2x + 5 = 0 by completing the square: (1) Move constant to right: x² + 2x = -5. (2) Take half of 2 to get 1, square it to get 1, add to both sides: x² + 2x + 1 = -5 + 1. (3) Left side is perfect square (x + 1)², right side simplifies to -4: (x + 1)² = -4. (4) Take square roots: x + 1 = ±√(-4) = ±2i. (5) Solve: x = -1 ± 2i. Done! Choice B correctly completes the square to get x = -1 ± 2i with accurate arithmetic and proper form, recognizing that √(-4) = 2i. Choice A forgets that we need the imaginary unit i when taking the square root of a negative number: √(-4) = 2i, not just 2. When q < 0 in (x - p)² = q, we get complex solutions! The ± is crucial: from (x - 3)² = 16, taking square roots gives x - 3 = ±4 (both +4 and -4), so x = 3 + 4 = 7 OR x = 3 - 4 = -1. Two solutions! Don't forget the ± and don't forget to split it into two separate solutions. When the right side is negative, remember to include i for the imaginary unit!

This question tests your ability to use completing the square to solve quadratic equations by transforming them into the form (x - p)² = q, which makes finding solutions straightforward by taking square roots. The completing-the-square process creates a perfect square trinomial on one side: for x² + bx, we add (b/2)²—half the middle coefficient, squared—to both sides. This turns x² + bx + (b/2)² into (x + b/2)², a perfect square that factors as a binomial squared. Then we solve by taking square roots. It's like setting up a problem in a form where the answer is obvious! Solving x² - 6x + 2 = 0 by completing the square: (1) Move constant to right: x² - 6x = -2. (2) Take half of -6 to get -3, square it to get 9, add to both sides: x² - 6x + 9 = -2 + 9. (3) Left side is perfect square (x - 3)², right side simplifies to 7: (x - 3)² = 7. (4) Take square roots: x - 3 = ±√7. (5) Solve: x = 3 ± √7. Done! Choice A correctly completes the square to get (x - 3)² = 7 and solves to get x = 3 ± √7 with accurate arithmetic and proper form. Choice B has the right perfect square form but makes a sign error: from (x - 3)², the sign inside is negative, but B uses (x + 3)² which would come from a positive coefficient. Watch those signs! When we have (x - 3)², it's equivalent to (x + (-3))², so the sign depends on b. The sign in the factored form is opposite to the p value. The (b/2)² trick: if you have x² + 6x, half of 6 is 3, and 3² = 9, so you add 9 to complete the square, getting (x + 3)². If you have x² - 10x, half of -10 is -5, and (-5)² = 25 (positive!), so add 25 to get (x - 5)². The sign of b affects the sign in the binomial, but (b/2)² is always positive. Practice this on a few examples and it becomes automatic!

This question tests your ability to use completing the square to solve quadratic equations by transforming them into the form (x - p)² = q, which makes finding solutions straightforward by taking square roots. The form (x - p)² = q tells us immediately: if q > 0, there are two real solutions (x = p ± √q); if q = 0, there's one solution (x = p); if q < 0, there are no real solutions (we'd need to take square root of negative). So completing the square not only solves the equation, it also reveals the nature of the solutions before we calculate them! Solving x² + 4x - 5 = 0 by completing the square: (1) Move constant to right: x² + 4x = 5. (2) Take half of 4 to get 2, square it to get 4, add to both sides: x² + 4x + 4 = 5 + 4. (3) Left side is perfect square (x + 2)², right side simplifies to 9: (x + 2)² = 9. (4) Take square roots: x + 2 = ±3. (5) Solve: x = -2 ± 3. Done! Choice C correctly completes the square to get (x + 2)² = 9 with accurate arithmetic and proper form. Choice D doesn't add (b/2)² to both sides—only adds to one side or miscalculates, leading to negative q incorrectly! Completing the square requires adding the perfect square term to BOTH sides to maintain equality. If you only add to the left, you've changed the equation and will get wrong solutions. Both sides, always! The ± is crucial: from (x - 3)² = 16, taking square roots gives x - 3 = ±4 (both +4 and -4), so x = 3 + 4 = 7 OR x = 3 - 4 = -1. Two solutions! Don't forget the ± and don't forget to split it into two separate solutions. Check both: 7 and -1 both satisfy the original equation? Yes!

This question tests your ability to use completing the square to solve quadratic equations by transforming them into the form (x - p)² = q, which makes finding solutions straightforward by taking square roots. Completing the square is a method that transforms any quadratic equation ax² + bx + c = 0 into an equivalent equation (x - p)² = q that has the same solutions but is much easier to solve: once in this form, we just take the square root of both sides (remembering ±!), giving x - p = ±√q, then solve for x = p ± √q. This method works for ALL quadratics, even ones that don't factor nicely! Solving x² + 6x + 5 = 0 by completing the square: (1) Move constant to right: x² + 6x = -5. (2) Take half of 6 to get 3, square it to get 9, add to both sides: x² + 6x + 9 = -5 + 9. (3) Left side is perfect square (x + 3)², right side simplifies to 4: (x + 3)² = 4. (4) Take square roots: x + 3 = ±2. (5) Solve: x = -3 ± 2, giving x = -3 + 2 = -1 or x = -3 - 2 = -5. Done! Choice A correctly completes the square to get (x + 3)² = 4 and solves to get x = -3 ± 2, which gives x = -1 or x = -5, with accurate arithmetic and proper form. Choice C makes an arithmetic error when simplifying the right side: -5 + 9 = 4, not 5, so we should get (x + 3)² = 4, not (x + 3)² = 5. Completing the square involves several arithmetic steps, so take your time and double-check! The complete completing-the-square procedure: (1) If a ≠ 1, divide everything by a first to get x² coefficient = 1, (2) Move the constant term to the right side, (3) Take half the x-coefficient, square it: (b/2)², (4) Add this to BOTH sides, (5) Factor left side as (x + b/2)² (it's a perfect square now!), (6) Simplify right side, (7) Take ± square root of both sides, (8) Solve for x. Follow these 8 steps systematically and you'll get the right answer every time!

This question tests your ability to use completing the square to solve quadratic equations by transforming them into the form (x - p)² = q, which makes finding solutions straightforward by taking square roots. The quadratic formula x = (-b ± √(b² - 4ac))/(2a) actually comes FROM completing the square on the general equation ax² + bx + c = 0! If you complete the square on this general form (treating a, b, c as constants), you derive the quadratic formula. So completing the square isn't just one method among many—it's the fundamental method that gives us the formula itself! Starting with ax² + bx + c = 0: (1) Divide by a: x² + (b/a)x + (c/a) = 0. (2) Move constant: x² + (b/a)x = -c/a. (3) Complete square, half of b/a is b/(2a), squared is b²/(4a²): x² + (b/a)x + b²/(4a²) = -c/a + b²/(4a²). (4) Left side: (x + b/(2a))². Right side: (b² - 4ac)/(4a²). (5) So (x + b/(2a))² = (b² - 4ac)/(4a²). (6) Square root: x + b/(2a) = ±√(b² - 4ac)/(2a). (7) Solve: x = -b/(2a) ± √(b² - 4ac)/(2a) = (-b ± √(b² - 4ac))/(2a). That's the quadratic formula! Choice C correctly derives the formula x = (-b ± √(b² - 4ac))/(2a) with accurate arithmetic and proper form. Choice A has the wrong sign under the radical: it should be b² - 4ac, not b² + 4ac. The discriminant b² - 4ac comes from combining the fractions on the right side when completing the square. This sign is crucial for determining the nature of solutions! Why this works: when you add (b/2)² to x² + bx, you're creating the expansion of (x + b/2)²: remember (a + b)² = a² + 2ab + b², so (x + b/2)² = x² + 2(x)(b/2) + (b/2)² = x² + bx + (b/2)². That's why adding (b/2)² makes a perfect square! It's not random—it completes the perfect square trinomial pattern.

This question tests your ability to use completing the square to solve quadratic equations by transforming them into the form (x - p)² = q, which makes finding solutions straightforward by taking square roots. Completing the square is a method that transforms any quadratic equation ax² + bx + c = 0 into an equivalent equation (x - p)² = q that has the same solutions but is much easier to solve: once in this form, we just take the square root of both sides (remembering ±!), giving x - p = ±√q, then solve for x = p ± √q. This method works for ALL quadratics, even ones that don't factor nicely! Solving x² - 6x + 2 = 0 by completing the square: (1) Move constant to right: x² - 6x = -2. (2) Take half of -6 to get -3, square it to get 9, add to both sides: x² - 6x + 9 = -2 + 9. (3) Left side is perfect square (x - 3)², right side simplifies to 7: (x - 3)² = 7. (4) Take square roots: x - 3 = ±√7. (5) Solve: x = 3 ± √7. Done! Choice A correctly completes the square to get (x - 3)² = 7 and solves to get x = 3 ± √7 with accurate arithmetic and proper form. Choice C makes an arithmetic error when simplifying the right side: -2 + 9 = 7, not 11, so we should get (x - 3)² = 7, not (x - 3)² = 11. Always double-check your arithmetic when combining terms on the right side after adding (b/2)² to both sides! Why this works: when you add (b/2)² to x² + bx, you're creating the expansion of (x + b/2)²: remember (a + b)² = a² + 2ab + b², so (x + b/2)² = x² + 2(x)(b/2) + (b/2)² = x² + bx + (b/2)². That's why adding (b/2)² makes a perfect square! It's not random—it completes the perfect square trinomial pattern.

This question tests your ability to use completing the square to solve quadratic equations by transforming them into the form (x - p)² = q, which makes finding solutions straightforward by taking square roots. The quadratic formula x = (-b ± √(b² - 4ac))/(2a) actually comes FROM completing the square on the general equation ax² + bx + c = 0! If you complete the square on this general form (treating a, b, c as constants), you derive the quadratic formula. So completing the square isn't just one method among many—it's the fundamental method that gives us the formula itself! Starting with ax² + bx + c = 0: (1) Divide by a: x² + (b/a)x + (c/a) = 0. (2) Move constant: x² + (b/a)x = -c/a. (3) Complete square, half of b/a is b/(2a), squared is b²/(4a²): x² + (b/a)x + b²/(4a²) = -c/a + b²/(4a²). (4) Left side: (x + b/(2a))². Right side: (b² - 4ac)/(4a²). (5) So (x + b/(2a))² = (b² - 4ac)/(4a²). (6) Square root: x + b/(2a) = ±√(b² - 4ac)/(2a). (7) Solve: x = -b/(2a) ± √(b² - 4ac)/(2a) = (-b ± √(b² - 4ac))/(2a). That's the quadratic formula! Choice C correctly derives the formula with accurate arithmetic and proper form. Choice A makes an arithmetic error when combining fractions in derivation: the right side is (b² - 4ac)/(4a²), not +4ac; completing the square involves several arithmetic steps, especially when deriving the formula with a, b, c. Take your time and double-check fraction arithmetic! To verify your completing-the-square work: expand your (x - p)² + k form back out using FOIL, and you should get back to your original quadratic. If you don't, there's an error. Also, solve the original equation using the quadratic formula—you should get the same solutions. These double-checks catch mistakes and build confidence!

This question tests your ability to use completing the square to solve quadratic equations by transforming them into the form (x - p)² = q, which makes finding solutions straightforward by taking square roots. Completing the square is a method that transforms any quadratic equation ax² + bx + c = 0 into an equivalent equation (x - p)² = q that has the same solutions but is much easier to solve: once in this form, we just take the square root of both sides (remembering ±!), giving x - p = ±√q, then solve for x = p ± √q. This method works for ALL quadratics, even ones that don't factor nicely! When a ≠ 1, like 2x² + 12x + 10 = 0: (1) First divide everything by 2: x² + 6x + 5 = 0. (2) Now complete the square on this simpler equation: x² + 6x = -5, add (6/2)² = 9 to both sides: x² + 6x + 9 = -5 + 9 = 4. (3) Factor left: (x + 3)² = 4. (4) Solve: x + 3 = ±2, so x = -3 ± 2, giving x = -1 or x = -5. Dividing by a first makes the completing square much cleaner! Choice C correctly solves to get x = -1 or x = -5 with accurate arithmetic and proper form. Choice A forgets to simplify after taking the square root: from (x + 3)² = 4, we get x + 3 = ±2 (not ±√4), since √4 = 2. Always simplify radicals: √4 = 2, √9 = 3, √16 = 4, etc. Don't leave them as √4 when they simplify to whole numbers! To verify your completing-the-square work: expand your (x - p)² + k form back out using FOIL, and you should get back to your original quadratic. If you don't, there's an error. Also, solve the original equation using the quadratic formula—you should get the same solutions. These double-checks catch mistakes and build confidence!