Question

NUMBER THEORY Use a quadratic equation to find two real numbers whose sum is 5 and whose product is -14, or show that no such numbers exist.

Short answer

The numbers are -2 and 7.

Step-by-step solution

Step-1 – Given

The sum = 5 and product = -14.

Step-2 – To determine

Use a quadratic equation to find two real numbers whose sum is 5 and whose product is -14, or show that no such numbers exist.

Step-3 – Calculation

Let the numbers are x and y.

So, the sum is $x+y=5$and product is $xy=-14$.

From $x+y=5$we get, $y=5-x$

Then we plug it in $xy=-14$and solve.

$\begin{array}{l}xy=-14\\ x(5-x)=-14\\ 5x-x^2=-14\\ x^2-5x-14=0\\ x^2-7x+2x-14=0\\ x(x-7)+2(x-7)=0\\ (x-7)(x+2)=0\\ x=7,-2\end{array}$

For ,$y=5-x=5-7=-2$

For , .$y=5-x=5-(-2)=5+2=7$

So, the numbers are -2 and 7.