Quadratic Equations
Example 1
The solutions of a quadratic equation are 4 and 1. Use their sum and
product to find a quadratic equation with those solutions.
Solution
For a quadratic equation, ax^{2} + bx + c = 0, the sum of the solutions is
The sum of the given solutions is
We have
So, if a = 1, then b = 3.
The product of the solutions is
The product of the given solutions is
We have
So, if a = 1, then c = 4.
For the quadratic equation, ax^{2} + bx + c = 0, let a
= 1, b = 3, and c = 4.

ax^{2} + bx + c
= 0 
Substitute these values in the equation.
Simplify. 
1x^{2} + (3)x  4 = 0
x^{2}  3x  4 = 0 
The equation, x^{2}  3x  4 = 0 is a quadratic equation
with solutions 4 and 1.Note:
x^{2}  3x  4 = 0
Any nonzero multiple of this equation
also has solutions 4 and 1.
Here are some examples:
2x^{2}  6x  8 = 0
3x^{2}  9x  12 = 0
Example 2
The quadratic equation 2x^{2}  7x + c = 0 has discriminant 9.
What are the solutions of the equation?
Solution
First, weâ€™ll find the value of c. Then weâ€™ll solve the equation.
To find c, we use the discriminant, b^{2}  4ac. For the given equation, a
= 2 and b = 7.
We also know the discriminant is 9.
Substitute a = 2 and b = 7.
Simplify.
Subtract 49 from both sides. Divide both sides by 8.
Now we know that c = 5. 
b^{2}  4ac = 9
(7)^{2}  4(2)c = 9
49  8c = 9
8c = 40 c = 5 
We use the quadratic formula to solve
the equation 2x^{2}  7x + c = 0. 

Substitute a = 2, b = 7, and c = 5. 

Simplify.


Simplify the radicand.


Simplify the square root.


Simplify. 

Thus, if the discriminant of 2x^{2}  7x + c = 0 is 9,
the solutions of the equation are
and 1.
