1­8 solve quadratic equations by using the quadratic formula and the discriminate ­ with notes.noteb
September 16, 2016
OBJECTIVE: Use the Quadratic Formula and the Discriminant to solve Quadratic equations.
WARM UP
1. Write 15x2 + 6x = 14x2 – 12 in standard form.
ANSWER
x2 + 6x +12 = 0
2. Evaluate b2 – 4ac when a = 3, b = –6, and c = 5. ANSWER
–24
3. A student is solving an equation by completing the
square. Write the step in the solution that appears
just before “(x – 3) = 5.”
ANSWER
(x – 3)2 = 25
1­8 solve quadratic equations by using the quadratic formula and the discriminate ­ with notes.noteb
September 16, 2016
OBJECTIVE: Use the Quadratic Formula and the Discriminant to solve Quadratic equations.
quadratic formula
can be used to solve all quadratic equations
Solve an equation with two REAL solutions
STEPS
1. Write original equation in standard form.
2. Identify a, b, and c.
3. Write Quadratic Formula.
4. Substitute values for variables in formula.
5. Simplify.
1­8 solve quadratic equations by using the quadratic formula and the discriminate ­ with notes.noteb
September 16, 2016
OBJECTIVE: Use the Quadratic Formula and the Discriminant to solve Quadratic equations.
Solve an equation with two real solutions
EXAMPLE:
2
Solve x
+ 3x = 2
x2 + 3x – 2 = 0
a = 1, b = 3, c = –2
The solutions are
ANSWER
CHECK
and
Use graphing calculator to check solutons
1­8 solve quadratic equations by using the quadratic formula and the discriminate ­ with notes.noteb
September 16, 2016
OBJECTIVE: Use the Quadratic Formula and the Discriminant to solve Quadratic equations.
Solve an equation with one real solution
EXAMPLE
2
Solve 25x – 18x = 12x – 9.
Write in standard form.
Identify a, b, and c.
write quadratic formula
substitute values for a, b, and c
simplify
a = 25, b = –30, c = 9
ANSWER
CHECK
The solution is
Use graphing calculator to check the solution.
1­8 solve quadratic equations by using the quadratic formula and the discriminate ­ with notes.noteb
September 16, 2016
OBJECTIVE: Use the Quadratic Formula and the Discriminant to solve Quadratic equations.
YOU TRY
1.
2.
x2 = 6x – 4
4x2 – 10x = 2x – 9
1­8 solve quadratic equations by using the quadratic formula and the discriminate ­ with notes.noteb
September 16, 2016
OBJECTIVE: Use the Quadratic Formula and the Discriminant to solve Quadratic equations.
Solve an equation with imaginary solutions
EXAMPLE
2
Solve –x + 4x = 5
Write in standard form
Identify a, b, and c.
a = –1, b = 4, c = –5
Substitute and simplify.
Rewrite using the imaginary unit i.
Simplify.
ANSWER
CHECK
The solutions are 2 + i and 2 – i.
Use graphing calculator to check solutions.CHECK
1­8 solve quadratic equations by using the quadratic formula and the discriminate ­ with notes.noteb
September 16, 2016
OBJECTIVE: Use the Quadratic Formula and the Discriminant to solve Quadratic equations.
YOU TRY
ANSWER
3. 7x – 5x2 – 4 = 2x + 3
1­8 solve quadratic equations by using the quadratic formula and the discriminate ­ with notes.noteb
September 16, 2016
OBJECTIVE: Use the Quadratic Formula and the Discriminant to solve Quadratic equations.
Use the discriminant
CONSIDER
2
Suppose the radicand, b
­ 4ac, is negative.
What do you know about the solution(s)?
2
Suppose the radicand, b
­ 4ac, is zero.
What do you know about the solution(s)?
2
Suppose the radicand, b
­ 4ac, is positive.
What do you know about the solution(s)?
discriminant
a relatively simple expression that determines the nature of the roots of a given equation or function. POSITIVE ­ Two real roots/solutions
ZERO ­ One real root/solution
NEGATIVE ­ Two imaginary roots/solutions
1­8 solve quadratic equations by using the quadratic formula and the discriminate ­ with notes.noteb
September 16, 2016
OBJECTIVE: Use the Quadratic Formula and the Discriminant to solve Quadratic equations.
Find the discriminant of the quadratic equation.
Give the number and type of solutions of the equation.
USE
x2 – 8x + 17 = 0
a. (–8)2 – 4(1)(17) = – 4
b. x2 – 8x + 16 = 0
(–8)2 – 4(1)(16) = 0
c. Two imaginary: 4 + i
One real: 4
x2 – 8x + 15 = 0
(–8)2 – 4(1)(15) = 4
Two real: 3,5
1­8 solve quadratic equations by using the quadratic formula and the discriminate ­ with notes.noteb
September 16, 2016
OBJECTIVE: Use the Quadratic Formula and the Discriminant to solve Quadratic equations.
HOMEWORK
Textbook page 62, column 2
page 63, #'s 52‐54