Methods for Solving Quadratic Equations
Quadratics equations are of the form ax2bxc 0, where a0 Quadratics may have two, one, or zero real solutions.
1. FACTORING
Set the equation equal to zero. If the quadratic side is factorable, factor, then set each factor equal to zero.
Example: x2 5x6
Move all terms to one side x25x60
Factor (x3)(x2)0
Set each factor to zero and solve x30 x20 2
3
x
x
2. PRINCIPLE OF SQUARE ROOTS
If the quadratic equation involves a SQUARE and a CONSTANT (no first degree term), position the square on one side and the constant on the other side. Then take the square root of both sides.
(Remember, you cannot take the square root of a negative number, so if this process leads to taking the square root of a negative number, there are no real solutions.)
Example 1: x2 160
Move the constant to the right side x2 16 Take the square root of both sides x2 16
4 4
,
4
which means x and x x
Example 2: 2(x3)2140
Move the constant to the other side 2(x3)2 14
Isolate the square (x3)2 7 (divide both sides by 2) Take the square root of both sides (x3)2 7
x3 7
Solve for x x 3 7
This represents the exact answer.
Decimal approximations can be found using a calculator.
3. COMPLETING THE SQUARE
If the quadratic equation is of the form ax2bxc0, where a0 and the quadratic expression is not factorable, try completing the square.
Example: x26x110
**Important: If a 1, divide all terms by “a” before proceeding to the next steps.
Move the constant to the right side x26x _____ 11 _____
and supply a blank on each side Find half of b, which means
2
b: 3 2 6
Square half of b: : 3 9
2
2 2
b Add
2
2
b
to both sides of the equation x2 6x9119 Factor the quadratic side (x3)(x3)20
(which is a perfect square because you just made it that way!) Then write in perfect square form (x3)2 20
Take the square root of both sides (x3)2 20 20 3
x Solve for x
5 2 3
20 3
x
radical the
Simplify x
This represents the exact answer.
Decimal approximations can be found using a calculator: 1.472 and -7.472
4. QUADRATIC FORMULA
Any quadratic equation of the form ax2bxc0, where a 0 can be solved for both real and imaginary solutions using the quadratic formula:
a ac b
x b
2
2 4
Example: x26x110 (a1, b6, c11) Substitute values into the quadratic formula:
answer EXACT
simplified final
the is This x
x
radical the
simplify x
x x
5 2 2 3
5 4 6
2 80 6 2
44 36 6 )
1 ( 2
) 11 )(
1 ( 4 6
6 2
