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:

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 \(x\):

$$x - 2 = 0 \implies x = 2$$

$$x - 5 = 0 \implies x = 5$$

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>