Find Two numbers whose Sum is 27 and Product is 182
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Updated Jun 14, 2024
Find Two numbers whose Sum is 27 and Product is 182
To find two numbers whose sum is 27 and product is 182, we can set up and solve a quadratic equation.
Let the first number be x.
Therefore, the second number will be 27−x.
The product of these two numbers is given as 182.
So, the equation based on their product is
x(27−x) = 182
Expanding the product, we get:
27x − x² = 182
Rearrange this to form a standard quadratic equation
x² − 27x + 182 = 0
We solve this quadratic equation using factorization
Factorize the quadratic equation
x² − 13x − 14x + 182 = 0
(x−13)(x−14) = 0
Set each factor to zero and solve for x
x − 13 = 0 or x − 14 = 0
x = 13 or x = 14
If the first number is 13, then the second number is
27 − 13 = 14
If the first number is 14, then the second number is
27 − 14 = 13
The two numbers whose sum is 27 and product is 182 are 13 and 14
How to Solve Quadratic Equations?
Quadratic equations are a fundamental part of algebra and are widely used in various fields of mathematics and science. A quadratic equation is a second-degree polynomial equation in the form ax²+bx+c=0, where a, b, and c are constants, and x represents an unknown variable.
Methods of Solving:
- Factoring: This method involves expressing the quadratic equation as a product of its factors and then setting each factor to zero to solve for x.
- Completing the Square: This method transforms the quadratic equation into a perfect square trinomial, making it easier to solve.
- Quadratic Formula: This formula provides the solution to any quadratic equation.
- Graphing: This method involves plotting the quadratic equation on a graph and identifying the points where the curve intersects the x-axis.
Discriminant:
The discriminant, denoted as Δ and calculated as b²−4ac, determines the nature of the roots of the quadratic equation.
If Δ>0, there are two distinct real roots.
If Δ=0, there is one real root (a repeated root).
If Δ<0, there are two complex roots.
Understanding quadratic equations is crucial for solving complex mathematical problems and for applications in various scientific and engineering fields.
