1. Quadratic Equations

2. Solving Quadratic Equations by the Square Root Property

3. Martin-Gay, Developmental Mathematics
Square Root Property
We previously have used factoring to solve quadratic equations.
This chapter will introduce additional methods for solving quadratic equations.
Square Root Property
If b is a real number and a2 = b, then
Martin-Gay, Developmental Mathematics

4. Martin-Gay, Developmental Mathematics
Square Root Property
Example
Solve x2 = 49
Solve 2x2 = 4
x2 = 2
Solve (y – 3)2 = 4
y = 3 ± 2
y = 1 or 5
Martin-Gay, Developmental Mathematics

5. Martin-Gay, Developmental Mathematics
Square Root Property
Example
Solve x2 + 4 = 0
x2 = −4
There is no real solution because the square root of −4 is not a real number.
Martin-Gay, Developmental Mathematics

6. Martin-Gay, Developmental Mathematics
Square Root Property
Example
Solve (x + 2)2 = 25
x = −2 ± 5
x = −2 + 5 or x = −2 – 5
x = 3 or x = −7
Martin-Gay, Developmental Mathematics

7. Martin-Gay, Developmental Mathematics
Square Root Property
Example
Solve (3x – 17)2 = 28
3x – 17 =
Martin-Gay, Developmental Mathematics

8. Solving Quadratic Equations by the Quadratic Formula

9. Martin-Gay, Developmental Mathematics
The Quadratic Formula
Another technique for solving quadratic equations is to use the quadratic formula.
The formula is derived from completing the square of a general quadratic equation.
Martin-Gay, Developmental Mathematics

10. Martin-Gay, Developmental Mathematics
The Quadratic Formula
A quadratic equation written in standard form, ax2 + bx + c = 0, has the solutions.
Martin-Gay, Developmental Mathematics

11. Solving Quadratic Equations
Steps in Solving Quadratic Equations
If the equation is in the form (ax+b)2 = c, use the square root property to solve.
If not solved in step 1, write the equation in standard form.
Try to solve by factoring.
If you haven’t solved it yet, use the quadratic formula.
Martin-Gay, Developmental Mathematics

12. Martin-Gay, Developmental Mathematics
The Quadratic Formula
Example
Solve 11n2 – 9n = 1 by the quadratic formula.
11n2 – 9n – 1 = 0, so
a = 11, b = -9, c = -1
Martin-Gay, Developmental Mathematics

13. Martin-Gay, Developmental Mathematics
The Quadratic Formula
Example
Solve x2 + x – = 0 by the quadratic formula.
x2 + 8x – 20 = (multiply both sides by 8)
a = 1, b = 8, c = -20
Martin-Gay, Developmental Mathematics

14. Martin-Gay, Developmental Mathematics
Solving Equations
Example
Solve the following quadratic equation.
Martin-Gay, Developmental Mathematics

15. Martin-Gay, Developmental Mathematics
The Quadratic Formula
Example
Solve x(x + 6) = -30 by the quadratic formula.
x2 + 6x + 30 = 0
a = 1, b = 6, c = 30
So there is no real solution.
Martin-Gay, Developmental Mathematics

16. Martin-Gay, Developmental Mathematics
The Discriminant
The expression under the radical sign in the formula (b2 – 4ac) is called the discriminant.
The discriminant will take on a value that is positive, 0, or negative.
The value of the discriminant indicates two distinct real solutions, one real solution, or no real solutions, respectively.
Martin-Gay, Developmental Mathematics

17. Martin-Gay, Developmental Mathematics
The Discriminant
Example
Use the discriminant to determine the number and type of solutions for the following equation.
5 – 4x + 12x2 = 0
a = 12, b = –4, and c = 5
b2 – 4ac = (–4)2 – 4(12)(5)
= 16 – 240
= –224
There are no real solutions.
Martin-Gay, Developmental Mathematics

18. Martin-Gay, Developmental Mathematics
Solving Equations
Example
Solve 3x = x2 + 1.
0 = x2 – 3x + 1
Let a = 1, b = -3, c = 1
Martin-Gay, Developmental Mathematics