Solve for x.
$2 x^2-7 x+3=0$
Answer (2)
Factor the quadratic equation 2 x 2 − 7 x + 3 = 0 by finding two numbers that multiply to 6 and add up to -7.
Rewrite the equation as 2 x 2 − 6 x − x + 3 = 0 and factor by grouping: 2 x ( x − 3 ) − 1 ( x − 3 ) = 0 .
Obtain the factored form ( 2 x − 1 ) ( x − 3 ) = 0 .
Solve for x by setting each factor to zero: x = 2 1 and x = 3 . The solutions are x = 2 1 , 3 .
Explanation
Understanding the Problem We are given the quadratic equation 2 x 2 − 7 x + 3 = 0 . Our goal is to find the values of x that satisfy this equation. We can solve this using the quadratic formula or by factoring. Let's try factoring first.
Factoring the Quadratic To factor the quadratic equation 2 x 2 − 7 x + 3 = 0 , we look for two numbers that multiply to 2 × 3 = 6 and add up to − 7 . These numbers are − 6 and − 1 . So we can rewrite the middle term as − 6 x − x .
Factoring by Grouping Now we rewrite the equation as 2 x 2 − 6 x − x + 3 = 0 . We can factor by grouping: 2 x ( x − 3 ) − 1 ( x − 3 ) = 0 .
Factored Form This gives us ( 2 x − 1 ) ( x − 3 ) = 0 .
Solving for x Now we set each factor equal to zero and solve for x :
2 x − 1 = 0 ⇒ 2 x = 1 ⇒ x = 2 1
x − 3 = 0 ⇒ x = 3
Final Answer The solutions to the quadratic equation 2 x 2 − 7 x + 3 = 0 are x = 2 1 and x = 3 .
Examples
Quadratic equations are used in various real-life scenarios, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its perimeter and area, or modeling the growth of a population. For example, if you want to build a rectangular garden with an area of 6 square meters and a perimeter of 14 meters, you can use a quadratic equation to find the length and width of the garden. Understanding how to solve quadratic equations helps in optimizing designs and predicting outcomes in many practical situations.
To solve the quadratic equation 2 x 2 − 7 x + 3 = 0 , we can factor it into ( 2 x − 1 ) ( x − 3 ) = 0 . The solutions to the equation are x = 2 1 and x = 3 .
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