Solve $z^2-2 z-35=0$ using the zero product property
A) $z=-1,-3$
B) $z=-3,5$
C) $z=7,-5$
D) $z=-7,-5$
Answer (1)
Factor the quadratic expression: z 2 − 2 z − 35 = ( z + 5 ) ( z − 7 ) .
Apply the zero product property: ( z + 5 ) = 0 or ( z − 7 ) = 0 .
Solve for z : z = − 5 or z = 7 .
The solutions are z = 7 , − 5 .
Explanation
Understanding the Problem We are given the quadratic equation z 2 − 2 z − 35 = 0 . Our goal is to solve this equation using the zero product property. This means we need to factor the quadratic expression into two binomials.
Finding the Factors To factor the quadratic expression z 2 − 2 z − 35 , we need to find two numbers that multiply to -35 and add up to -2. Let's call these numbers a and b . So, we need to find a and b such that a × b = − 35 and a + b = − 2 .
Identifying the Correct Pair The factors of -35 are: (1, -35), (-1, 35), (5, -7), and (-5, 7). Among these pairs, the pair (-5, 7) satisfies the condition that their sum is -2, since − 5 + 7 = 2 . However, we need the sum to be -2, so we consider the pair (5, -7). Indeed, 5 + ( − 7 ) = − 2 and 5 × ( − 7 ) = − 35 . Therefore, a = 5 and b = − 7 .
Factoring the Quadratic Now we can rewrite the quadratic expression in factored form as ( z + 5 ) ( z − 7 ) = 0 .
Applying the Zero Product Property Using the zero product property, we set each factor equal to zero: z + 5 = 0 or z − 7 = 0 .
Solving for z Solving for z in each case, we get: z = − 5 or z = 7 .
Final Answer Therefore, the solutions to the quadratic equation z 2 − 2 z − 35 = 0 are z = − 5 and z = 7 .
Examples
Solving quadratic equations by factoring is a fundamental skill in algebra and has many real-world applications. For example, if you are designing a rectangular garden with a specific area and a constraint on the difference between the length and width, you might need to solve a quadratic equation to find the dimensions of the garden. Similarly, in physics, projectile motion problems often involve solving quadratic equations to determine the time it takes for an object to reach a certain height or distance.
