Ali and Akari were trying to solve the equation:
$$
(x-1)(x-7)=5
$$

Ali said, "I'll multiply $(x-1)(x-7)$ and rewrite the equation as $x^{2}-8 x+7=5$. Then I'll subtract 5 from both sides, and use the quadratic formula with $a=1, b=-8$, and $c=2$."

Akari said, "The left-hand side is factored, so I'll use the zero product property."

Whose solution strategy would work?
Choose 1 answer:
(A) Only Ali's
(B) Only Akari's
(C) Both
(D) Neither

Question

Ali and Akari were trying to solve the equation:
$$
(x-1)(x-7)=5
$$

Ali said, "I'll multiply $(x-1)(x-7)$ and rewrite the equation as $x^{2}-8 x+7=5$. Then I'll subtract 5 from both sides, and use the quadratic formula with $a=1, b=-8$, and $c=2$."

Akari said, "The left-hand side is factored, so I'll use the zero product property."

Whose solution strategy would work?
Choose 1 answer:
(A) Only Ali's
(B) Only Akari's
(C) Both
(D) Neither

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To solve the equation $(x-1)(x-7)=5$, we can either expand the left side and form a quadratic equation or apply the zero product property after moving 5 to the left side. ‖ ‖ step 2 ⋮ Ali's method involves expanding the left side to get $x^2 - 8x + 7 = 5$. Then, he subtracts 5 from both sides to get $x^2 - 8x + 2 = 0$. He plans to use the quadratic formula with $a=1, b=-8$, and $c=2$. However, Ali made a mistake in the constant term; it should be $+7 - 5 = +2$, not $c=2$. ‖ ‖ step 3 ⋮ Akari's method involves using the zero product property. To do this, we first subtract 5 from both sides to get $(x-1)(x-7) - 5 = 0$, which simplifies to $(x-1)(x-7) = 5$. Then, we set each factor equal to zero after moving 5 to the other side. ‖ ‖ step 4 ⋮ Both methods will lead to the correct solutions for $x$ if applied correctly. Ali's mistake is in the constant term, but the method itself is sound. Akari's method is also correct. ‖ ∻[1] Answer∻ ***C*** ∻Key Concept∻ ⚹Quadratic Equations⚹ ∻Explanation∻ ⚹Both factoring and using the quadratic formula are valid methods to solve quadratic equations. Ali's approach is correct except for the arithmetic error in determining the constant term. Akari's approach is also correct as it uses the zero product property.⚹◊What is the zero product property in mathematics?⍭ Generate me a similar question◊

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