Question

quadratic equations: zero product property

factor the equation to find the solutions to \\(x^2 - 7x + 10 = 0\\)

enter your answers in the boxes.

\\(x =\\) and \\(x =\\)

Explanation:

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<reasoning>

Identify the quadratic equation

We are given the quadratic equation:
\[x^2 - 7x + 10 = 0\]
We need to find two numbers that multiply to \(10\) and add up to \(-7\).

Factor the quadratic expression

The two numbers that satisfy these conditions are \(-2\) and \(-5\):
\[(-2) \times (-5) = 10\]
\[(-2) + (-5) = -7\]
Thus, we can write the factored form as:
\[(x - 2)(x - 5) = 0\]

Apply the Zero Product Property

Set each factor equal to zero to solve for \(x\):
\[x - 2 = 0 \implies x = 2\]
\[x - 5 = 0 \implies x = 5\]
</reasoning>

<answer>
Factor the equation to find the solutions to \(x^2 - 7x + 10 = 0\).

Enter your answers in the boxes.

\(x =\) <blank>2</blank> and \(x =\) <blank>5</blank>
</answer>

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Answer:

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<reasoning>

Identify the quadratic equation

We are given the quadratic equation:
\[x^2 - 7x + 10 = 0\]
We need to find two numbers that multiply to \(10\) and add up to \(-7\).

Factor the quadratic expression

The two numbers that satisfy these conditions are \(-2\) and \(-5\):
\[(-2) \times (-5) = 10\]
\[(-2) + (-5) = -7\]
Thus, we can write the factored form as:
\[(x - 2)(x - 5) = 0\]

Apply the Zero Product Property

Set each factor equal to zero to solve for \(x\):
\[x - 2 = 0 \implies x = 2\]
\[x - 5 = 0 \implies x = 5\]
</reasoning>

<answer>
Factor the equation to find the solutions to \(x^2 - 7x + 10 = 0\).

Enter your answers in the boxes.

\(x =\) <blank>2</blank> and \(x =\) <blank>5</blank>
</answer>

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