Mid-Chapter Quiz
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Begin by writing two equations, one for the sum of the numbers and one for their product.
No such numbers exist.
Let x and y be the desired real numbers. Using the given information, we can write two equations, one for the sum of the numbers and one for their product. Sum:& x+ y=7 Product:& x y=15 We can use these equations to form a system of equations. x+y=7 & (I) xy=15 & (II) We can solve this system using the substitution method. First, let's isolate y in Equation (I).
(II): y= 7-x
(II): Distribute x
(II): LHS-15=RHS-15
(II): Commutative Property of Addition
The quadratic equation in our system can be written as - x^2+7x-15=0. Our next step is to solve the quadratic equation. We can do this by graphing. Let's begin by finding the axis of symmetry using the formula x=- b2a. In our equation, a=-1 and b=7.
Next, we will make a table of values using x values around the axis of symmetry, which is x= 72 or, in decimal form, x=3.5.
| x | - x^2+7x-15 | y=- x^2+7x-15 |
|---|---|---|
| 1 | - ( 1)^2+7( 1)-15 | - 9 |
| 2 | - ( 2)^2+7( 2)-15 | - 5 |
| 3.5 | - ( 3.5)^2+7( 3.5)-15 | - 2.75 |
| 5 | - ( 5)^2+7( 5)-15 | - 5 |
| 6 | - ( 6)^2+7( 6)-15 | - 9 |
Now, we will plot the points and graph the equation that passes through them.
The graph does not cross the x-axis, so our system has no real solutions. This means that no real numbers exist that satisfy the given conditions.