Solve the quadratic by factoring: [tex]3 x^2-11 x+10=0[/tex]
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Answer (1)
Factor the quadratic expression: 3 x 2 − 11 x + 10 = ( 3 x − 5 ) ( x − 2 ) .
Set each factor to zero: 3 x − 5 = 0 or x − 2 = 0 .
Solve for x : x = 3 5 or x = 2 .
The solutions are x = 2 , 3 5 .
Explanation
Understanding the Problem We are given the quadratic equation 3 x 2 − 11 x + 10 = 0 and asked to solve it by factoring. Our goal is to find the values of x that satisfy this equation.
Factoring the Quadratic To factor the quadratic expression 3 x 2 − 11 x + 10 , we look for two numbers that multiply to 3 × 10 = 30 and add up to − 11 . These numbers are − 5 and − 6 . We can rewrite the middle term using these numbers: 3 x 2 − 11 x + 10 = 3 x 2 − 5 x − 6 x + 10
Factoring by Grouping Now, we factor by grouping: 3 x 2 − 5 x − 6 x + 10 = x ( 3 x − 5 ) − 2 ( 3 x − 5 ) We can factor out the common term ( 3 x − 5 ) : x ( 3 x − 5 ) − 2 ( 3 x − 5 ) = ( 3 x − 5 ) ( x − 2 )
Solving for x Now we set each factor equal to zero and solve for x :
3 x − 5 = 0 or x − 2 = 0
Solving 3 x − 5 = 0 :
3 x = 5
x = 3 5
Solving x − 2 = 0 :
x = 2
Final Answer Therefore, the solutions to the quadratic equation 3 x 2 − 11 x + 10 = 0 are x = 3 5 and x = 2 .
Examples
Quadratic equations are useful in many real-world applications, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its area and a relationship between its sides, or modeling the growth of a population. For example, if you are designing a bridge, you might use a quadratic equation to model the curve of an arch.
