Algebra 1 Chapter 8B Note-Taking Guide
Solving Quadratics
8.5
Name_______________________
Per ___ Date _________________
Every quadratic function has a related quadratic equation. A quadratic equation is an equation that
can be written in the standard form _________________, where a, b, and c are real numbers and a ≠
0.
When writing a quadratic function as its related quadratic equation, you replace ________________.
y = ax2 + bx + c
0 = ax2 + bx + c
_____________________
One way to solve a quadratic equation in standard form is to _______________________________
and find the x-values where __________. In other words, find the ________________ of the related
function. Recall that a quadratic function may have ___________ zeros.
8.5
Using a provided graph to solve each quadratic equation.
1.) Solve
2 x 2  4 x  6  0
2.) Solve
x2  4x  7  0
3.) Solve
 x2  6x  9  0
4.) Solve
x2  8x  7  0
8.5
Creating your own graph to solve a quadratic equation
5.) Solve
2 x 2  12 x  18  0
6.) Solve
x 2  8 x  16  0
7.) Solve
x 2  6 x  10  0
8.) Solve
 x2  4  0
8.6
You have solved quadratic equations by graphing. Another method used to solve quadratic equations
is to factor and use the Zero Product Property.
8.6
Use the Zero Product Property to solve the equation.
A.) (x – 7)(x + 2) = 0
B.) (x + 4)(x – 3) = 0
C.) x(x - 5) = 0
D.) x(2x – 3) = 0
F.) (5x + 4)(x – 3) = 0
8.6
E.) (3x – 7)(2x + 1) = 0
Solve each equation:
G.) x2 – 6x + 8 = 0
H.) x2 + 4x – 21 = 0
I.) 3x2 – 4x + 1 = 0
J.) 2x2 + 27x – 14 = 0
8.7
Solving quadratics by using square roots
Some quadratic equations cannot be easily solved by factoring. Square roots can be used to solve
some of these quadratic equations. Recall from Lesson 1-5 that every positive real number has two
square roots, one _______________ and one _________________.
Square root property:
To solve a quadratic equation in the form x2 = a, take the square root of both sides. So the answer will
be
8.7
8.7
**As long as a is positive
Solve the following equations:
a.) x 2  25
b.) x 2  169
c.) x 2  9
If necessary, use inverse operations to isolate the squared part of a quadratic equation before taking
the square root of both sides
d.) x 2  4  12
e.) x 2  20  101
f.) 2 x 2  32
g.)
x2
 32
2
j.)  x  5   16
2
h.) 16 x 2  49  0
i.) 100 x 2  9  0
k.)  2 x  1  25
2
8.7
8.7
8.7
8.8
When solving quadratic equations by using square roots, you may need to find the square root of a
number that is not a perfect square. In this case, the answer is an irrational number. You can
approximate the solutions using a calculator.
l.) 3(x - 1)2 = 15
m.) x2 - 10 = 25
Review:
1. x2 – 195 = 1
2. 4x2 – 18 = –9
3. (x + 7)2 = 81
4. 0 = –5x2 + 225
Solving by completing the square
In the previous lesson, you solved quadratic equations by isolating x2 and then using square roots. This
method works if the quadratic equation, when written in standard form, is a perfect square.
When a trinomial is a perfect square, there is a relationship between the coefficient of the x-term and
the constant term.
x2 + 6x + 9
x2 – 8x + 16
These can be factored into a perfect square trinomial
(x + 3)2
(x - 4)2
8.8
An expression in the form x2 + bx is not a perfect square. However, you can use the relationship shown
above to add a term to x2 + bx to form a trinomial that is a perfect square. This is called completing
the square.
Complete the square to form a perfect square trinomial. Then factor the trinomial.
A.) x2 + 2x + ____
B.) x2 - 6x + _____
C.) x2 + 12x + ____
D.) x2 - 24x + _____
8.8
To solve a quadratic equation in the form x2 + bx = c, first complete the square of x2 + bx. Then you
can solve using square roots.
 Divide “b” value by 2, then square the answer
 Add that to both sides
 Factor the perfect square trinomial
 Square root both sides
 Solve
Solve by completing the square.
E.) x2 + 16x = –15
F.) x2 + 10x = –9
G.) x2 – 4x = 6
H.) x2 – 8x – 5 = 0
I.) –3x2 + 12x – 15 = 0
8.9
J.) 5x2 + 20x = 35
In the previous lesson, you completed the square to solve quadratic equations. If you complete the
2
square of ax + bx + c = 0, you can derive the ______________________________.
The Quadratic Formula is the only method that can be used to solve ______________quadratic
equation.
The Quadratic Formula
The solutions of ax 2  bx  c  0 (where a ≠ 0) can be found by the formula
x
8.9
Use the quadratic formula to solve each:
2
a.) 6x + 5x – 4 = 0
2
b.) x - x = 20
2
c.) –3x + 5x + 2 = 0
8.9
Many quadratic equations can be solved by graphing, factoring, taking the square root, or completing
the square. Some ___________ be solved by any of these methods, but you can ___________ use
the Quadratic Formula to solve any quadratic equation.
2
a.) x + 3x + 7 = 0
2
b.) 2x – 8x + 1 = 0
8.9
If the quadratic equation is in standard form, the discriminant of a quadratic equation is
_______________, the part of the equation under the radical sign. Recall that quadratic equations
can have two, one, or no real solutions. You can determine the number of solutions of a quadratic
equation by evaluating its discriminant.
Discriminant is
Number of Real
Solutions
Positive
Zero
Negative
Determine the number of real solutions for each equation by using the discriminant.
2
a.) 3x – 2x + 2 = 0
2
b.) 2x + 11x + 12 = 0
2
c.) x + 8x + 16 = 0
8.10 When solving a system with a quadratic and line, there are three possibilities:

_____________________________

_____________________________

_____________________________
To find a solution of a system by GRAPHING, graph
both equations, and then find their intersection.
y  x 1

2
 y  x  4x  5
 y  x2  2x  3

 y  x 1
8.10 When solving a system using substitution, replace the y in one equation with what y is equal to in the
second equation. Then you can factor, complete the square, or use the quadratic formula to solve the
system.
 y  2 x 2  3x  4

y  x  2
 y  3x 2  3x  1

 y  3 x  4