Use the zero product property to find the solutions to the equation [tex]$6 x^2-5 x=56$[/tex].
A. [tex]$x=-8$[/tex] or [tex]$x=\frac{7}{6}$[/tex]
B. [tex]$x=-\frac{1}{2}$[/tex] or [tex]$x=\frac{56}{3}$[/tex]
C. [tex]$x=-\frac{8}{3}$[/tex] or [tex]$x=\frac{7}{2}$[/tex]
D. [tex]$x=-\frac{3}{7}$[/tex] or [tex]$x=\frac{2}{3}$[/tex]
Answer (2)
Rewrite the given equation 6 x 2 − 5 x = 56 in standard form: 6 x 2 − 5 x − 56 = 0 .
Factor the quadratic expression: ( 3 x + 8 ) ( 2 x − 7 ) = 0 .
Apply the zero product property: 3 x + 8 = 0 or 2 x − 7 = 0 .
Solve for x : x = − 3 8 or x = 2 7 .
Explanation
Rewrite the Equation We are given the quadratic equation 6 x 2 − 5 x = 56 . Our goal is to find the solutions for x using the zero product property. This property states that if ab = 0 , then either a = 0 or b = 0 (or both). To use this property, we first need to rewrite the equation in the standard form of a quadratic equation, which is a x 2 + b x + c = 0 .
Standard Form To get the equation in standard form, we subtract 56 from both sides of the equation: 6 x 2 − 5 x − 56 = 0
Factoring Now we need to factor the quadratic expression 6 x 2 − 5 x − 56 . We are looking for two numbers that multiply to 6 × − 56 = − 336 and add up to − 5 . These numbers are − 21 and 16 . So we can rewrite the middle term as − 21 x + 16 x :
6 x 2 − 21 x + 16 x − 56 = 0
Factoring by Grouping Next, we factor by grouping. From the first two terms, we can factor out 3 x , and from the last two terms, we can factor out 8 :
3 x ( 2 x − 7 ) + 8 ( 2 x − 7 ) = 0
Complete Factoring Now we can factor out the common factor ( 2 x − 7 ) :
( 3 x + 8 ) ( 2 x − 7 ) = 0
Zero Product Property Now we apply the zero product property. We set each factor equal to zero and solve for x :
3 x + 8 = 0 or 2 x − 7 = 0
Solve First Equation Solving the first equation for x :
3 x = − 8 ⇒ x = − 3 8
Solve Second Equation Solving the second equation for x :
2 x = 7 ⇒ x = 2 7
Final Answer Therefore, the solutions to the equation 6 x 2 − 5 x = 56 are x = − 3 8 and x = 2 7 .
Examples
Understanding quadratic equations and the zero product property is crucial in various fields, such as physics and engineering. For instance, when calculating the trajectory of a projectile, you often end up with a quadratic equation that needs to be solved to find the time it takes for the projectile to hit the ground. Similarly, in electrical engineering, quadratic equations are used to analyze circuits and determine the values of components that satisfy certain conditions. Mastering these concepts provides a strong foundation for solving real-world problems in these disciplines. Imagine designing a bridge where the load distribution can be modeled by a quadratic equation; finding the roots helps determine critical stress points.
The solutions to the equation 6 x 2 − 5 x = 56 are x = − 3 8 and x = 2 7 . Using the zero product property after factoring the quadratic yields these solutions. Thus, the correct option is C.
;
