This question tests your ability to use completing the square to solve quadratic equations and understand how it leads to solutions, much like deriving the quadratic formula shows it's rooted in algebra. Completing the square transforms ax² + bx + c = 0 into (x - p)² = q, solved by x = p ± √q, working for all quadratics including those with real or complex roots; if q > 0, two real solutions; q = 0, one; q < 0, two complex. For x² + 8x + 7 = 0, move 7: x² + 8x = -7, add (8/2)² = 16: (x + 4)² = 9, then x + 4 = ±3, so x = -1 or -7—great job following the steps! Choice A correctly completes the square to (x + 4)² = 9 and finds both solutions accurately. Choice B mistakenly uses half of 8 as 8 instead of 4, leading to an incorrect perfect square term and wrong q value—remember, it's always half the coefficient squared! The systematic process is: move constant, add (b/2)² to both sides, factor, take ± square root, solve—practice this and you'll master it every time! Deriving the quadratic formula follows similarly on ax² + bx + c = 0, yielding x = [-b ± √(b² - 4ac)] / (2a), proving the formula's algebraic foundation.

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 $x = p \pm \sqrt{q}$. 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 $2x^2 + 12x - 10 = 0$ by completing the square: (1) Divide by 2: $x^2 + 6x - 5 = 0$. (2) Move constant: $x^2 + 6x = 5$. (3) Half of 6 is 3, square 9, add: $x^2 + 6x + 9 = 14$. (4) $(x + 3)^2 = 14$. (5) $x = -3 \pm \sqrt{14}$. 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 forgets to add the correct amount after dividing, leading to wrong $q=4$ instead of 14. 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 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 $x^2 + 10x - 3 = 0$ by completing the square: (1) Move constant: $x^2 + 10x = 3$. (2) Half of 10 is 5, square 25, add: $x^2 + 10x + 25 = 28$. (3) $(x + 5)^2 = 28$. (4) $x + 5 = \pm \sqrt{28} = \pm 2\sqrt{7}$. (5) $x = -5 \pm 2\sqrt{7}$. 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 D miscalculates q as 22 instead of 28 after adding 25 to 3. 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 $x = p \pm \sqrt{q}$. 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 $x^2 + 6x - 2 = 0$ by completing the square: (1) Move constant to right: $x^2 + 6x = 2$. (2) Take half of 6 to get 3, square it to get 9, add to both sides: $x^2 + 6x + 9 = 2 + 9 = 11$. (3) Factor left side as perfect square: $(x + 3)^2 = 11$. (4) Take square roots: $x + 3 = \pm \sqrt{11}$. (5) Solve: $x = -3 \pm \sqrt{11}$. 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 calculates (b/2)^2 incorrectly: half of 6 is 3, and 3 squared is 9, leading to +9 on right making 11, not 7. 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 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 ± 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 ax² + bx + c = 0 into the form (x - p)² = q, which is easy to solve by taking square roots: x - p = ±√q, giving x = p ± √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 ± i√|q|. The value of q immediately reveals the nature of solutions. To solve x² + 4x - 5 = 0 by completing the square: (1) Move constant: x² + 4x = 5. (2) Half of 4 is 2, square 4, add: x² + 4x + 4 = 9. (3) (x + 2)² = 9. (4) x + 2 = ±3. (5) x = -2 ± 3, so x = 1 or -5. The systematic process works every time! For deriving the quadratic formula, complete the square on ax² + 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 gets q=1 instead of 9, likely forgetting to add 4 to the right side correctly. 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 ± 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 ax² + bx + c = 0 treating a, b, c as constants: divide by a to get x² + (b/a)x + (c/a) = 0, then complete square on x² + (b/a)x by adding (b/(2a))², which gives (x + b/(2a))² = (b² - 4ac)/(4a²), then take square root and solve to get x = (-b ± √(b² - 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 $x^2 - 6x + 13 = 0$ by completing the square: (1) Move constant: $x^2 - 6x = -13$. (2) Half of -6 is -3, square 9, add: $x^2 - 6x + 9 = -4$. (3) $(x - 3)^2 = -4$. (4) $x - 3 = \pm 2i$. (5) $x = 3 \pm 2i$. 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 gets positive $q=4$, but actual $q=-4$ for complex roots. 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 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: $x - p = \pm \sqrt{q}$, 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!