Question

Find two numbers whose sum is 29 and product is 210

Answer 1

Need-to-know

Equation

An equation has unknowns. Certain values of the unknown satisfy the equation and it is called a solution.

Quadratic Equation

This problem has to do with the quadratic equation. The reason will be explained later.

Solution

Let the numbers be α and β.

Then we have,

$\begin{cases} & \alpha +\beta =29 \\ & \alpha \beta = 210\end{cases}$

We know an equation that has both numbers as a solution is,

$(x-\alpha )(x-\beta )=0$

$\longleftrightarrow x^2-(\alpha +\beta )x+\alpha \beta=0$

Then our equation becomes

$x^2-29x+210=0$

Which has solutions as,

$\alpha ,\beta =\dfrac{29\pm\sqrt{29^2-4\times 210} }{2}$

Let's put aside and find $29^2$. We know this holds for every number.

$(a-b)^2=a^2-2ab+b^2$

So let's put our number, $30-1$ inside this equation.

$(30-1)^2=(30)^2-2\times 30+1^2$

                $=900-60+1$

                $=841$

Returning to our problem, let's start calculating the solutions.

$\alpha ,\beta =\dfrac{29\pm\sqrt{29^2-4\times 210} }{2}$

        $=\dfrac{29\pm \sqrt{841-840} }{2}$

        $=\dfrac{29\pm 1}{2}=14,15$

Both are $\boxed{14}$ or $\boxed{15}$. So, this is the desired pair of numbers.

Answer 2

Step-by-step explanation:

Given :-

Find two numbers whose sum is 29 and product is 210

To Find :-

Find two numbers

Answer :-

$29x \: - \: {x}^{2} = 210 \\ \\ {x}^{2} - 29x \: + \: 210 = 0 \\ \\ {x}^{2} \: - \: 15x \: - \: 14x \: + \: 210 = 0 \\ \\ x(x \: - \: 15) \: - \: 14(x \: - \: 15) = 0 \\ \\ (x \: - \: 14) \: (x \: - 15) = 0 \\ \\ x = 14 \: and \: 15$

Sum, 14 + 15 = 29

Product, 14 × 15 = 210