Question

use the zero product property to find the solutions to the equation $x^2 - 13x + 30 = 0$.
\\(\circ\\) $x = -10$ or $x = -3$
\\(\circ\\) $x = -10$ or $x = 3$
\\(\circ\\) $x = -3$ or $x = 10$
\\(\circ\\) $x = 3$ or $x = 10$

Explanation:

Step1: Factor the quadratic equation

To solve \(x^{2}-13x + 30=0\) using the zero - product property, we first factor the quadratic trinomial. We need two numbers that multiply to \(30\) (the constant term) and add up to \(- 13\) (the coefficient of the \(x\) term). The numbers are \(-3\) and \(-10\) since \((-3)\times(-10) = 30\) and \((-3)+(-10)=-13\). So, we can factor the quadratic as \((x - 3)(x - 10)=0\).

Step2: Apply the zero - product property

The zero - product property states that if \(ab = 0\), then either \(a = 0\) or \(b = 0\) (or both). For the equation \((x - 3)(x - 10)=0\), we set each factor equal to zero:

• If \(x - 3=0\), then by adding \(3\) to both sides of the equation, we get \(x=3\).
• If \(x - 10=0\), then by adding \(10\) to both sides of the equation, we get \(x = 10\).

Answer:

\(x = 3\) or \(x = 10\) (corresponding to the option: \(x = 3\) or \(x = 10\))