Question

Find the two numbers that have a sum of 2 and a product of $\mathbf{-}\mathbf{15}$.

Short answer

The two numbers are $\mathbf{-}\mathbf{3}$and 5.

Step-by-step solution

Step 1. Define the standard form of the quadratic equation.

The standard form of the quadratic equation is given by

$a{x}^2+bx+c=0$Where, $a\ne 0$.

Step 2. Form a quadratic equation by using the question.

Let the first number $=x$

Since, the sum of the two numbers is 2.

So, the second number $=2-x$

Since, the product of the two numbers is $-15$.

Therefore,

role="math" localid="1647869395507" $$\begin{gathered} x\left(2-x\right)=-15 \\ 2x-x^2=-15 \\ {x}^2-2x-15=0 \end{gathered}$$

Step 3. Solve the equation x2−2x−15=0.

Split the mid-term to solve the equation $x^2-2x-15=0$.

role="math" localid="1647869538451" $$\begin{gathered} x^2-5x+3x-15=0 \\ x\left(x-5\right)+3\left(x-5\right)=0 \\ \left(x-5\right)\left(x+3\right)=0 \\ x=-3,5 \end{gathered}$$

Now at, $x=-3$

The second number $=2-\left(-3\right)=2+3=5$

Now at $x=5$

The second number $=2-5=-3$

So, the two numbers can be $-3$and 5.

Therefore, the two numbers are$-3$ and 5.