Download Lecture notes for Section 8.2

Int. Alg. Notes
Section 8.2
Page 1 of 4
Section 8.2: Solving Quadratic Equations by the Quadratic Formula
Big Idea: Another way to solve a quadratic equation ax 2  bx  c  0 is to plug its coefficients into the
quadratic formula:
The Quadratic Formula: x 
 b  b 2  4ac
2a
Big Skill: You should be able to solve a quadratic equation using the quadratic formula.
The Quadratic Formula
The solution(s) to the quadratic formula ax 2  bx  c  0 (for a  0) are given by the quadratic formula:
x
b  b 2  4ac
2a
Proof:
The quadratic formula is derived by completing the square on the standard from of a quadratic equation:
 Get the constant term on the right hand side of the equation.
ax 2  bx  c  0


ax 2  bx  c
Make sure the coefficient of the square term is 1.
ax 2  bx  c
ax 2  bx
c

a
a
b
c
x2  x  
a
a
Identify the coefficient of the linear term; multiply it by ½ and square the result.
2

b2
1 b




4a 2
2 a
Add that number to both sides of the equation.
b
c
x2  x  
a
a
2
b
b
c b2
x2  x  2    2
a
4a
a 4a
2
b
b
b2 c
2
x  x 2  2 
a
4a
4a a
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes

Section 8.2
Page 2 of 4
Write the resulting perfect square trinomial as the square of the binomial .
b
b2
b2 c
x2  x  2  2 
a
4a
4a a
2
b 
b 2 4ac

x





2a 
4a 2 4a 2

b  b 2  4ac

x  
2a 
4a 2

Use the square root property to solve the equation.
2

b 
b 2  4ac

x





2a 
4a 2

2
x
b
b 2  4ac

2a
2a
x
b
b 2  4ac

2a
2a
x

b  b 2  4ac
2a
b2 – 4ac is called the discriminant. The discriminant is important because it determines the nature of
the solutions (roots) of the quadratic equation.
Examples of the Nature of the Roots of a Quadratic Equation with Rational Coefficients:
1. If b2 – 4ac is positive and a perfect square, then there are two solutions that are real, rational, and
unequal. In this case, you can also solve the quadratic equation by factoring.
Example: 2 x 2  5 x  1.125  0
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 8.2
Page 3 of 4
2. If b2 – 4ac is positive but not a perfect square, then there are two solutions that are real, irrational, and
unequal.
Example: 1.5 x 2  7 x  3  0
3. If b2 – 4ac = 0, then there is just one solution (a repeated root) that is real and rational (or we can say
that the two solutions are equal). This case can also be solved by factoring.
Example: 9 x 2  24 x  16  0
4. If b2 – 4ac is negative, then there are two solutions that are complex and unequal.
Example: 4 x 2  2 x  9  0
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 8.2
Page 4 of 4
Practice:
5. Solve for k: 16k 
9
 24
k
6. The length of a tennis court is 12.8 m more than its width. If the area of a tennis court is 262 m2, what
are its dimensions?
7. The golden ratio is important in architecture and design because it is the foundation for the most
pleasing looking rectangles and linear proportions. The golden ratio is the ratio of the length to the
width of a rectangle such that when you remove a square that has side equal to the width of the
rectangle, the remaining rectangle has sides that also are in proportion to the golden ratio. Calculate the
golden ratio.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.