1. 10
Quadratic Equations

2. 10.3 Solving Quadratic Equations by the Quadratic Formula
Objectives
1. Identify the values of a, b, and c in a quadratic equation. 2. Use the quadratic formula to solve quadratic equations. 3. Solve quadratic equations with only one solution. 4. Solve quadratic equations with fractions.

3. Note
In ax2+bx+c=0, there is a restriction that a is not zero.
If it were, the equation would be linear, not quadratic.

4. Identify the Values of a, b, and c in Quadratic Equations
Example 1
Identify the values of the variables of a, b, and c, in each quadratic equation ax2 + bx + c = 0.
(a) –3x2 + 5 = 2x
Write the equation in standard form ax2 + bx + c=0.
–3x2 –2x + 5 = 0
a = –3, b = –2, and c = 5

5. Identify the Values of a, b, and c in Quadratic Equations
Example 1 (continued)
Identify the values of the variables of a, b, and c, in each quadratic equation ax2 + bx + c = 0.
(b) (x – 2)(x + 2) = 0
x2 – 4 = 0
Multiply using the FOIL method.
x2 + 0x – 4 = 0
The x-term is missing, so write the equation with 0 as the coefficient of the x-term.
a = 1, b = 0, and c = –4

6. Use the Quadratic Formula to Solve Quadratic Equations
Step 1 Transform so that the coefficient of x2 is equal to 1.
Standard form
Divide by a.
Step 2 Write so that the variable terms with x are alone on the left side.
Subtract .

7. Use the Quadratic Formula to Solve Quadratic Equations
Step 3 Add the square of half the coefficient of x to each
side, factor the left side, and combine terms on the
right.
Add
Factor; add on right.

8. Use the Quadratic Formula to Solve Quadratic Equations
Step 4 Use the square root property to complete the solution.

9. Use the Quadratic Formula to Solve Quadratic Equations
The solutions of the quadratic equation ax2 + bx + c, a ≠ 0, are
or in compact form,

10. Use the Quadratic Formula to Solve Quadratic Equations
CAUTION
Notice in the quadratic formula that the fraction bar is
under –b as well as the radical. Be sure to find the
values of first, and then divide those
results by the value of 2a.

11. Use the Quadratic Formula to Solve Quadratic Equations
Example 2 Solve 3x2 + 4x – 5 = 0.
In this equation, a = 3, b = 4, and c = –5.

12. Solve Quadratic Equations with Only One Solution
Example 4 Solve 16x2 – 24x + 9 = 0.
In this equation, a = 16, b = –24, and c = 9.
There is just one solution in the solution set,

13. Solve Quadratic Equations with Fractions
Example 5
Solve
Standard form
Clear fractions; multiply by the LCD, 4.
4x2 – 26x + 46 = 0
Multiply.
In this equation, a = 4, b = –26 , and c = 46.

14. Solve Quadratic Equations with Fractions
Example 5 (concluded)
Solve
is not a real number.
The solution set is