Published On:Thursday, 8 December 2011
Posted by Muhammad Atif Saeed
Solving Quadratic Equations
Solving Quadratic Equations
A quadratic equation is an equation that could be written as: ax2 + bx + c = 0
when a 0. There are three basic methods for solving quadratic equations: factoring, using the quadratic formula, and completing the square.
Factoring
To solve a quadratic equation by factoring,- Put all terms on one side of the equal sign, leaving zero on the other side.
- Factor.
- Set each factor equal to zero.
- Solve each of these equations.
- Check by inserting your answer in the original equation.
Example 1
Solve x2 – 6 x = 16.Following the steps,
x2 – 6 x = 16 becomes x2 – 6 x – 16 = 0
Factor.( x – 8)( x + 2) = 0
Setting each factor to zero, 
Example 2
Solve y2 = – 6 y – 5.Setting all terms equal to zero,
y2 + 6 y + 5 = 0
Factor.( y + 5)( y + 1) = 0
Setting each factor to 0, 
Example 3
Solve x2 – 16 = 0.Factor.
Example 4
Solve x2 + 6 x = 0.Factor.
Example 5
Solve 2 x2 + 2 x – 1 = x2 + 6 x – 5.First, simplify by putting all terms on one side and combining like terms.
The quadratic formula
Many quadratic equations cannot be solved by factoring. This is generally true when the roots, or answers, are not rational numbers. A second method of solving quadratic equations involves the use of the following formula:
ax2 + bx + c = 0
where a is the numeral that goes in front of x2, b is the numeral that goes in front of x, and c is the numeral with no variable next to it (a.k.a., “the constant”). When using the quadratic formula, you should be aware of three possibilities. These three possibilities are distinguished by a part of the formula called the discriminant. The discriminant is the value under the radical sign, b2 – 4 ac. A quadratic equation with real numbers as coefficients can have the following:
- Two different real roots if the discriminant b2 – 4 ac is a positive number.
- One real root if the discriminant b2 – 4 ac is equal to 0.
- No real root if the discriminant b2 – 4 ac is a negative number.
Example 6
Solve for x: x2 – 5 x = –6.Setting all terms equal to 0,
x2 – 5 x + 6 = 0
Then substitute 1 (which is understood to be in front of the x2), –5, and 6 for a, b, and c, respectively, in the quadratic formula and simplify. 
Example 7
Solve for y: y2 = –2y + 2.Setting all terms equal to 0,
y2 + 2 y – 2 = 0
Then substitute 1, 2, and –2 for a, b, and c, respectively, in the quadratic formula and simplify. 
Example 8
Solve for x: x2 + 2 x + 1 = 0.Substituting in the quadratic formula,
The quadratic formula can also be used to solve quadratic equations whose roots are imaginary numbers, that is, they have no solution in the real number system.
Example 9
Solve for x: x( x + 2) + 2 = 0, or x2 + 2 x + 2 = 0.Substituting in the quadratic formula,
But if you were to express the solution using imaginary numbers, the solutions would be
. Completing the square
A third method of solving quadratic equations that works with both real and imaginary roots is called completing the square.- Put the equation into the form ax2 + bx = – c.
- Make sure that a = 1 (if a ≠ 1, multiply through the equation by
before proceeding).
- Using the value of b from this new equation, add
to both sides of the equation to form a perfect square on the left side of the equation.
- Find the square root of both sides of the equation.
- Solve the resulting equation.
Example 10
Solve for x: x2 – 6 x + 5 = 0.Arrange in the form of
, or 9, to both sides to complete the square. 
x – 3 = ±2
Solve.
Example 11
Solve for y: y2+ 2 y – 4 = 0.Arrange in the form of
, or 1, to both sides to complete the square. 
Example 12
Solve for x: 2 x2 + 3 x + 2 = 0.Arrange in the form of
. 
or 
to both sides. 
