Solve by factorisation
[tex]2 x^2+3 x-20=0[/tex]
Answer (2)
To solve the quadratic equation 2 x 2 + 3 x − 20 = 0 , we factor it into ( 2 x − 5 ) ( x + 4 ) = 0 and find the solutions as x = 2 5 and x = − 4 .
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Factor the quadratic expression: Find two numbers that multiply to 2 × − 20 = − 40 and add up to 3 (which are 8 and − 5 ).
Rewrite the middle term: 2 x 2 + 8 x − 5 x − 20 = 0 .
Factor by grouping: 2 x ( x + 4 ) − 5 ( x + 4 ) = 0 , which simplifies to ( 2 x − 5 ) ( x + 4 ) = 0 .
Solve for x : 2 x − 5 = 0 gives x = 2 5 , and x + 4 = 0 gives x = − 4 . Thus, the solutions are x = 2 5 , − 4 .
Explanation
Understanding the Problem We are given the quadratic equation 2 x 2 + 3 x − 20 = 0 and asked to solve it by factorization. This means we need to rewrite the quadratic expression as a product of two linear expressions.
Finding the Right Numbers To factorize the quadratic expression 2 x 2 + 3 x − 20 , we look for two numbers whose product is 2 × − 20 = − 40 and whose sum is 3 . These numbers are 8 and − 5 .
Rewriting the Middle Term We rewrite the middle term using these numbers: 2 x 2 + 8 x − 5 x − 20 = 0 .
Factoring by Grouping Now, we factor by grouping: 2 x ( x + 4 ) − 5 ( x + 4 ) = 0 .
Factoring out the Common Term We factor out the common term ( x + 4 ) : ( 2 x − 5 ) ( x + 4 ) = 0 .
Setting Factors to Zero Setting each factor equal to zero, we have 2 x − 5 = 0 or x + 4 = 0 .
Solving for x (First Solution) Solving 2 x − 5 = 0 for x , we get 2 x = 5 , so x = 2 5 .
Solving for x (Second Solution) Solving x + 4 = 0 for x , we get x = − 4 .
Final Answer Therefore, the solutions to the quadratic equation 2 x 2 + 3 x − 20 = 0 are x = 2 5 and x = − 4 .
Examples
Quadratic equations are used in various real-life situations, 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 growth and decay processes. For example, if you're launching a rocket, you can use a quadratic equation to predict its path, ensuring it reaches its target safely. Understanding how to solve quadratic equations helps in making accurate predictions and informed decisions in these scenarios.
