Form a quadratic equation whose roots are 3 and 4.
Answer (2)
To form a quadratic equation with roots 3 and 4, you can use the factored form (x - 3)(x - 4) = 0, which expands to x² - 7x + 12 = 0. This is the required quadratic equation. Therefore, the quadratic equation is x² - 7x + 12 = 0.
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Let the roots be α = 3 and β = 4 .
Form the equation ( x − α ) ( x − β ) = 0 .
Substitute the values: ( x − 3 ) ( x − 4 ) = 0 .
Expand and simplify: x 2 − 7 x + 12 = 0 . The required quadratic equation is x 2 − 7 x + 12 = 0 .
Explanation
Understanding the Problem We are given that the roots of the quadratic equation are 3 and 4. Our goal is to find the quadratic equation that has these roots.
Forming the Equation Let the roots be α = 3 and β = 4 . A quadratic equation with roots α and β can be written in the form ( x − α ) ( x − β ) = 0 . This is because when x = α or x = β , the equation becomes zero, satisfying the condition for roots.
Substituting the Roots Substitute the values of α and β into the equation: ( x − 3 ) ( x − 4 ) = 0 .
Expanding the Equation Expand the equation: x 2 − 4 x − 3 x + 12 = 0 . We multiply each term in the first parenthesis by each term in the second parenthesis.
Simplifying the Equation Simplify the equation: x 2 − 7 x + 12 = 0 . We combine the like terms − 4 x and − 3 x to get − 7 x .
Final Answer Therefore, the required quadratic equation is x 2 − 7 x + 12 = 0 .
Examples
Quadratic equations are used in various real-life applications, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its area and perimeter, and modeling curves in engineering and physics. For instance, if you want to build a rectangular garden with an area of 12 square meters and you know one side is 3 meters, you can use a quadratic equation to find the length of the other side. In this case, the equation would be 3 x = 12 , which simplifies to x = 4 . The quadratic equation we found, x 2 − 7 x + 12 = 0 , represents a scenario where the roots (3 and 4) could be the dimensions of a rectangle with a specific area and perimeter.
