This question tests your ability to derive the quadratic formula by completing the square on the general form, showing it's not magic but systematic algebra. Completing the square transforms any quadratic $ax^2 + bx + c = 0$ into the form $(x - p)^2 = q$, which is easy to solve by taking square roots: $x - p = \pm \sqrt{q}$, giving $x = p \pm \sqrt{q}$. This method works for ALL quadratics—even those that don't factor and even those with complex solutions! If $q$ is positive, you get two real solutions; if $q = 0$, one solution; if $q$ negative, two complex solutions $x = p \pm i \sqrt{|q|}$. The value of $q$ immediately reveals the nature of solutions. To derive the quadratic formula, complete the square on $ax^2 + bx + c = 0$ treating $a$, $b$, $c$ as constants: divide by $a$ to get $x^2 + (b/a)x + (c/a) = 0$, then move $c/a$: $x^2 + (b/a)x = -c/a$, add $(b/(2a))^2$ to both: $(x + b/(2a))^2 = -c/a + (b/(2a))^2 = (b^2 - 4ac)/(4a^2)$, then take square roots: $x + b/(2a) = \pm \sqrt{(b^2 - 4ac)/(4a^2)} = \pm \sqrt{b^2 - 4ac}/(2a)$, so $x = -b/(2a) \pm \sqrt{b^2 - 4ac}/(2a) = [-b \pm \sqrt{b^2 - 4ac}]/(2a)$. This derivation shows completing the square is the foundation! Choice C correctly derives the formula with the proper signs and discriminant. Choice A has a sign error in the discriminant, using $+4ac$ instead of $-4ac$, which would mess up the nature of solutions. The systematic completing-the-square procedure: (1) If $a \ne 1$, divide everything by $a$ first to get $x^2$ coefficient $= 1$, (2) Move constant to right, (3) Take half the x-coefficient, square it: $(b/2)^2$, (4) Add to BOTH sides, (5) Factor left as $(x + b/2)^2$, (6) Simplify right, (7) Take $\pm$ square root, (8) Solve for $x$. To derive the quadratic formula, apply these steps symbolically as above—track the algebra carefully, and you'll see how the formula emerges every time! Amazing insight, well done!

This question tests your ability to use completing the square to solve quadratic equations and, importantly, to derive the quadratic formula—showing it's not magic but comes from systematic algebra. Completing the square transforms any quadratic $ax^2 + bx + c = 0$ into the form $(x - p)^2 = q$, which is easy to solve by taking square roots: $x - p = \pm \sqrt{q}$, giving $x = p \pm \sqrt{q}$. To solve $x^2 - 8x + 5 = 0$ by completing the square: (1) Move constant to right: $x^2 - 8x = -5$. (2) Take half of -8 to get -4, square it to get 16, add to both sides: $x^2 - 8x + 16 = -5 + 16 = 11$. (3) Factor left side: $(x - 4)^2 = 11$. (4) Take square roots: $x - 4 = \pm \sqrt{11}$. (5) Solve: $x = 4 \pm \sqrt{11}$. Choice A correctly completes the square to get $(x - 4)^2 = 11$ and solves for both solutions. Choice B incorrectly factors as $(x + 4)^2$ instead of $(x - 4)^2$—when the x coefficient is -8, half is -4, giving $(x - 4)^2$, not $(x + 4)^2$! The systematic procedure ensures you get the correct factored form by carefully tracking the sign of b/2.

This question tests your ability to use completing the square to solve quadratic equations and, importantly, to derive the quadratic formula—showing it's not magic but comes from systematic algebra. Completing the square transforms any quadratic $ax^2 + bx + c = 0$ into the form $(x - p)^2 = q$, which is easy to solve by taking square roots: $x - p = \pm \sqrt{q}$, giving $x = p \pm \sqrt{q}$. This method works for ALL quadratics—even those that don't factor and even those with complex solutions! If $q$ is positive, you get two real solutions; if $q = 0$, one solution; if $q$ negative, two complex solutions $x = p \pm i \sqrt{|q|}$. The value of $q$ immediately reveals the nature of solutions. To solve $x^2 + 8x + 7 = 0$ by completing the square: (1) Move constant to right: $x^2 + 8x = -7$. (2) Take half of 8 to get 4, square it to get 16, add to both sides: $x^2 + 8x + 16 = -7 + 16 = 9$. (3) Factor left side as perfect square: $(x + 4)^2 = 9$. (4) Take square roots: $x + 4 = \pm 3$. (5) Solve: $x = -4 \pm 3$, so $x = -1$ or $x = -7$. The systematic process works every time! For deriving the quadratic formula, complete the square on $ax^2 + bx + c = 0$ treating $a$, $b$, $c$ as constants, following the same steps but keeping everything symbolic. Choice A correctly completes the square to get the proper form and solves to find both solutions accurately. Choice B calculates $(b/2)^2$ incorrectly: half of 8 is 4, and 4 squared is 16, not leading to $(x+8)^2=57$. This perfect square term is crucial—getting it wrong throws off everything! The systematic completing-the-square procedure: (1) If $a \neq 1$, divide everything by $a$ first to get $x^2$ coefficient = 1, (2) Move constant to right, (3) Take half the x-coefficient, square it: $(b/2)^2$, (4) Add to BOTH sides, (5) Factor left as $(x + b/2)^2$, (6) Simplify right, (7) Take $\pm$ square root, (8) Solve for $x$. Follow these 8 steps and you'll get correct answer every time! To derive the quadratic formula, complete the square on $ax^2 + bx + c = 0$ treating $a$, $b$, $c$ as constants: divide by $a$ to get $x^2 + (b/a)x + (c/a) = 0$, then complete square on $x^2 + (b/a)x$ by adding $(b/(2a))^2$, which gives $(x + b/(2a))^2 = (b^2 - 4ac)/(4a^2)$, then take square root and solve to get $x = (-b \pm \sqrt{b^2 - 4ac}) / (2a)$. This derivation shows completing the square is the fundamental method underlying the quadratic formula!

This question tests your ability to use completing the square to solve quadratic equations without factoring, even if it seems tempting. Completing the square transforms any quadratic $ax^2 + bx + c = 0$ into the form $(x - p)^2 = q$, which is easy to solve by taking square roots: $x - p = \pm \sqrt{q}$, giving $x = p \pm \sqrt{q}$. This method works for ALL quadratics—even those that don't factor and even those with complex solutions! If q is positive, you get two real solutions; if q = 0, one solution; if q negative, two complex solutions $x = p \pm i \sqrt{|q|}$. The value of q immediately reveals the nature of solutions. To solve $x^2 + 12x + 20 = 0$: (1) Move constant: $x^2 + 12x = -20$. (2) Half of 12 is 6, square 36, add both sides: $x^2 + 12x + 36 = -20 + 36 = 16$. (3) $(x + 6)^2 = 16$. (4) $x + 6 = \pm 4$. (5) $x = -6 \pm 4$, so x = -2 or -10. Perfect execution! Choice A correctly completes the square to get the proper form and solves for both solutions. Choice D flips the sign of p, likely by using half of -12 instead, which reverses the center. The systematic completing-the-square procedure: (1) If a ≠ 1, divide everything by a first to get x² coefficient = 1, (2) Move constant to right, (3) Take half the x-coefficient, square it: $(b/2)^2$, (4) Add to BOTH sides, (5) Factor left as $(x + b/2)^2$, (6) Simplify right, (7) Take $\pm$ square root, (8) Solve for x. Follow these 8 steps and you'll get the correct answer every time! You're mastering this!

This question tests your ability to use completing the square to solve quadratic equations and, importantly, to derive the quadratic formula—showing it's not magic but comes from systematic algebra. Completing the square transforms any quadratic $ax^2 + bx + c = 0$ into the form $(x - p)^2 = q$, which is easy to solve by taking square roots: $x - p = \pm \sqrt{q}$, giving $x = p \pm \sqrt{q}$. This method works for ALL quadratics—even those that don't factor and even those with complex solutions! If $q$ is positive, you get two real solutions; if $q = 0$, one solution; if $q$ negative, two complex solutions $x = p \pm i\sqrt{|q|}$. The value of $q$ immediately reveals the nature of solutions. To derive the quadratic formula, complete the square on $ax^2 + bx + c = 0$ treating $a$, $b$, $c$ as constants: divide by $a$ to get $x^2 + (b/a)x + (c/a) = 0$, move constant: $x^2 + (b/a)x = -c/a$, add $(b/(2a))^2$ to both: $(x + b/(2a))^2 = (b^2/(4a^2)) - (c/a) = (b^2 - 4ac)/(4a^2)$, take $\pm$ square root: $x + b/(2a) = \pm \sqrt{(b^2 - 4ac)/(4a^2)} = \pm(\sqrt{b^2 - 4ac})/(2a)$, so $x = -b/(2a) \pm(\sqrt{b^2 - 4ac})/(2a) = [-b \pm \sqrt{b^2 - 4ac}] / (2a)$. The systematic process works every time! Choice C correctly derives the quadratic formula through completing the square on the general form. Choice A makes an error in the derivation of the quadratic formula during the completing square process with the general form, using $+4ac$ instead of $-4ac$ in the discriminant. When working with $a$, $b$, $c$ as variables rather than numbers, track each step carefully—algebraic errors compound quickly in derivations! The systematic completing-the-square procedure: (1) If $a \neq 1$, divide everything by $a$ first to get $x^2$ coefficient $= 1$, (2) Move constant to right, (3) Take half the x-coefficient, square it: $(b/2)^2$, (4) Add to BOTH sides, (5) Factor left as $(x + b/2)^2$, (6) Simplify right, (7) Take $\pm$ square root, (8) Solve for $x$. Follow these 8 steps and you'll get correct answer every time! To derive the quadratic formula, complete the square on $ax^2 + bx + c = 0$ treating $a$, $b$, $c$ as constants: divide by $a$ to get $x^2 + (b/a)x + (c/a) = 0$, then complete square on $x^2 + (b/a)x$ by adding $(b/(2a))^2$, which gives $(x + b/(2a))^2 = (b^2 - 4ac)/(4a^2)$, then take square root and solve to get $x = (-b \pm \sqrt{b^2 - 4ac}) / (2a)$. This derivation shows completing the square is the fundamental method underlying the quadratic formula!

This question tests your ability to use completing the square to solve quadratic equations and, importantly, to derive the quadratic formula—showing it's not magic but comes from systematic algebra. Completing the square transforms any quadratic $ax^2 + bx + c = 0$ into the form $(x - p)^2 = q$, which is easy to solve by taking square roots: $x - p = \pm \sqrt{q}$, giving $x = p \pm \sqrt{q}$. This method works for ALL quadratics—even those that don't factor and even those with complex solutions! If q is positive, you get two real solutions; if q = 0, one solution; if q negative, two complex solutions $x = p \pm i\sqrt{|q|}$. The value of q immediately reveals the nature of solutions. To solve $x^2 + 2x + 5 = 0$ by completing the square: (1) Move constant: $x^2 + 2x = -5$. (2) Half of 2 is 1, square 1, add: $x^2 + 2x + 1 = -4$. (3) $(x + 1)^2 = -4$. (4) $x + 1 = \pm 2i$. (5) $x = -1 \pm 2i$. The systematic process works every time! For deriving the quadratic formula, complete the square on $ax^2 + bx + c = 0$ treating a, b, c as constants, following the same steps but keeping everything symbolic. Choice B correctly completes the square to get the proper form and solves to find both solutions accurately. Choice A incorrectly gets positive q=4, but it's negative for complex roots. This perfect square term is crucial—getting it wrong throws off everything! The systematic completing-the-square procedure: (1) If a ≠ 1, divide everything by a first to get x² coefficient = 1, (2) Move constant to right, (3) Take half the x-coefficient, square it: $(b/2)^2$, (4) Add to BOTH sides, (5) Factor left as $(x + b/2)^2$, (6) Simplify right, (7) Take $\pm$ square root, (8) Solve for x. Follow these 8 steps and you'll get correct answer every time! To derive the quadratic formula, complete the square on $ax^2 + bx + c = 0$ treating a, b, c as constants: divide by a to get $x^2 + (b/a)x + (c/a) = 0$, then complete square on $x^2 + (b/a)x$ by adding $(b/(2a))^2$, which gives $(x + b/(2a))^2 = (b^2 - 4ac)/(4a^2)$, then take square root and solve to get $x = (-b \pm \sqrt{b^2 - 4ac}) / (2a)$. This derivation shows completing the square is the fundamental method underlying the quadratic formula!

This question tests your ability to use completing the square to solve quadratic equations and, importantly, to derive the quadratic formula—showing it's not magic but comes from systematic algebra. Completing the square transforms any quadratic $ax^2 + bx + c = 0$ into the form $(x - p)^2 = q$, which is easy to solve by taking square roots; if q is negative, you get complex solutions $x = p \pm i\sqrt{|q|}$. To solve $x^2 - 6x + 13 = 0$: (1) Move constant: $x^2 - 6x = -13$. (2) Take half of -6 to get -3, square it to get 9, add to both sides: $x^2 - 6x + 9 = -13 + 9 = -4$. (3) Factor: $(x - 3)^2 = -4$. (4) Take square roots: $x - 3 = \pm \sqrt{-4} = \pm 2i$. (5) Solve: $x = 3 \pm 2i$. Choice A correctly completes the square to get $(x - 3)^2 = -4$ and recognizes that $\sqrt{-4} = 2i$, giving complex solutions. Choice B incorrectly factors as $(x + 3)^2$ instead of $(x - 3)^2$—when the coefficient of x is -6, half is -3, so we get $(x - 3)^2$, not $(x + 3)^2$! The systematic procedure reveals when solutions are complex: whenever the right side after completing the square is negative.

This question tests your ability to use completing the square to solve quadratic equations and, importantly, to derive the quadratic formula—showing it's not magic but comes from systematic algebra. Completing the square transforms any quadratic $ax^2 + bx + c = 0$ into the form $(x - p)^2 = q$, which is easy to solve by taking square roots: $x - p = \pm \sqrt{q}$, giving $x = p \pm \sqrt{q}$. This method works for ALL quadratics—even those that don't factor and even those with complex solutions! If q is positive, you get two real solutions; if q = 0, one solution; if q negative, two complex solutions $x = p \pm i\sqrt{|q|}$. The value of q immediately reveals the nature of solutions. To solve $x^2 - 12x + 20 = 0$ by completing the square: (1) Move constant: $x^2 - 12x = -20$. (2) Half of -12 is -6, square 36, add: $x^2 - 12x + 36 = 16$. (3) $(x - 6)^2 = 16$. (4) $x - 6 = \pm 4$. (5) $x = 6 \pm 4$, so x = 10 or 2. The systematic process works every time! For deriving the quadratic formula, complete the square on $ax^2 + bx + c = 0$ treating a, b, c as constants, following the same steps but keeping everything symbolic. Choice A correctly completes the square to get the proper form and solves to find both solutions accurately. Choice C has the sign wrong for p, using +6 instead of -6 for negative b term. This perfect square term is crucial—getting it wrong throws off everything! The systematic completing-the-square procedure: (1) If a ≠ 1, divide everything by a first to get x² coefficient = 1, (2) Move constant to right, (3) Take half the x-coefficient, square it: $(b/2)^2$, (4) Add to BOTH sides, (5) Factor left as $(x + b/2)^2$, (6) Simplify right, (7) Take $\pm$ square root, (8) Solve for x. Follow these 8 steps and you'll get correct answer every time! To derive the quadratic formula, complete the square on $ax^2 + bx + c = 0$ treating a, b, c as constants: divide by a to get $x^2 + (b/a)x + (c/a) = 0$, then complete square on $x^2 + (b/a)x$ by adding $(b/(2a))^2$, which gives $(x + b/(2a))^2 = (b^2 - 4ac)/(4a^2)$, then take square root and solve to get $x = (-b \pm \sqrt{b^2 - 4ac}) / (2a)$. This derivation shows completing the square is the fundamental method underlying the quadratic formula!

This question tests your ability to use completing the square to solve quadratic equations and, importantly, to derive the quadratic formula—showing it's not magic but comes from systematic algebra. Completing the square transforms any quadratic $ax^2 + bx + c = 0$ into the form $(x - p)^2 = q$, which is easy to solve by taking square roots: $x - p = \pm \sqrt{q}$, giving $x = p \pm \sqrt{q}$. To solve $x^2 + 12x + 20 = 0$ by completing the square: (1) Move constant to right: $x^2 + 12x = -20$. (2) Take half of 12 to get 6, square it to get 36, add to both sides: $x^2 + 12x + 36 = -20 + 36 = 16$. (3) Factor left side: $(x + 6)^2 = 16$. (4) Take square roots: $x + 6 = \pm 4$. (5) Solve: $x = -6 \pm 4$, so $x = -2$ or $x = -10$. Choice A correctly completes the square to get $(x + 6)^2 = 16$ and recognizes that $\sqrt{16} = 4$. Choice B incorrectly calculates $-20 + 36 = 56$ instead of 16—always double-check your arithmetic! The systematic procedure gives you $x = -6 \pm 4$, which means $x = -2$ or $x = -10$.

This question tests your ability to use completing the square to solve quadratic equations that factor nicely, but remember to use the method as instructed. Completing the square transforms any quadratic $ax^2 + bx + c = 0$ into the form $(x - p)^2 = q$, which is easy to solve by taking square roots: $x - p = \pm \sqrt{q}$, giving $x = p \pm \sqrt{q}$. This method works for ALL quadratics—even those that don't factor and even those with complex solutions! If q is positive, you get two real solutions; if q = 0, one solution; if q negative, two complex solutions $x = p \pm i\sqrt{|q|}$. The value of q immediately reveals the nature of solutions. To solve $x^2 + 4x - 5 = 0$: (1) Move constant: $x^2 + 4x = 5$. (2) Half of 4 is 2, square 4, add both sides: $x^2 + 4x + 4 = 5 + 4 = 9$. (3) $(x + 2)^2 = 9$. (4) $x + 2 = \pm 3$. (5) $x = -2 \pm 3$, so $x = 1$ or $-5$. You're nailing the steps! Choice A correctly completes the square to get the proper form and finds both integer solutions. Choice B miscalculates q as 1, perhaps by forgetting to add the square term to the right side properly. The systematic completing-the-square procedure: (1) If a ≠ 1, divide everything by a first to get x² coefficient = 1, (2) Move constant to right, (3) Take half the x-coefficient, square it: $(b/2)^2$, (4) Add to BOTH sides, (5) Factor left as $(x + b/2)^2$, (6) Simplify right, (7) Take $\pm$ square root, (8) Solve for x. Follow these 8 steps and you'll get the correct answer every time! Keep up the great progress!