Name: ____________________________________
Date: __________________
Solving Quadratic Equations Using Quadratic Functions
Algebra 1
Today we will be using quadratic functions to assist us in solving quadratic equations.
Definition: A quadratic equation is an equation that can be written in the standard form
ax 2  bx  c  0, where a, b, and c are real numbers and a  0 . The solutions of a quadratic equation
are called its roots (zeros).
Notice the similarity between a quadratic equation and a quadratic function.
Quadratic Function:
y  ax 2  bx  c
0  ax 2  bx  c
Quadratic Equation:
SOLVING A QUADRATIC EQUATION BY USING ITS RELATED QUADRATIC FUNCTION
Graphically
Graph the related quadratic
function and find the x-intercepts
(because this is where y = 0).
Using a Table
Create a table for the
quadratic function and find
the x values for which y = 0.
Exercise #1: The graph of the function y  x 2  3x  4 is given. According to the graph, the roots of
y
the equation x 2  3 x  4  0 are
(1)
(2)
(3) 1.5,  6
1, 4
4
(4) 1, 2
x
Exercise #2: A table of values is given below for the function y  x 2  2 x  15. Use the table to
determine the values of x for which x 2  2 x  15  0. Check using the STORE command.
x
–5
–4
–3
–2
–1
0
1
2
3
4
5
6
7
y
20
9
0
–7
–12
–15
–16
–15
–12
–7
0
9
20
Algebra 1, Unit #5 - Quadratic Functions – L4
The Arlington Algebra Project, LaGrangeville, NY 12540
Exercise #3: Find the zeros of the function y  x 2  3x  54 numerically by using a table of values.
Create the table with your graphing calculator.
Exercise #4: The graph of a particular function of the form y  ax 2  bx  c, where a, b, and c are real
numbers, is shown below. Use the graph to answer the following questions.
y
x
(a) Is the numerical value of a positive or negative? Justify.
(b) State the numerical value of c.
(c) State all solutions to each equation below:
(i) ax 2  bx  c  0.
(ii) ax 2  bx  c  3.
Exercise #5: Using the accompanying grids, sketch graphs of functions of the form y  ax 2  bx  c
that satisfy the given criteria.
(a) ax 2  bx  c  0 has two
roots; a  0
(b) ax 2  bx  c  0 has exactly
one root; a  0
y
y
y
x
Algebra 1, Unit #5 – Quadratic Functions – L4
The Arlington Algebra Project, LaGrangeville, NY 12540
(c) ax 2  bx  c  0 has
no roots; a  0
x
x
Fact: A quadratic equation can have 0, 1, or 2 solutions.
Exercise #6: Determine the roots for each quadratic equation given below by using your graphing
calculator. If the equation has no roots, then so state.
(a) x 2  x  6  0
(b)  x 2  6 x  9  0
(c) x 2  4 x  6  0
Exercise #7: Which of the following functions has two zeros?
(1) y  x 2  7 x  14
(2) y  x 2  10 x  25
(3) y   x 2  5 x  6
(4) y  x 2  4
Exercise #8: If one x-intercept of the graph of a quadratic function is 4 and the axis of symmetry
has equation x  3 , then what is the other x-intercept?
Exercise #9: Which table below illustrates a quadratic function with a maximum value of zero?
(1)
(3)
(2)
(4)
Algebra 1, Unit #5 – Quadratic Functions – L4
The Arlington Algebra Project, LaGrangeville, NY 12540
Name: ____________________________________
Date: __________________
Solving Quadratic Equations Using Quadratic Functions
Algebra 1 Homework
Skills
1. The graph of y  x 2  6 x  8 is shown. The roots of the equation x 2  6 x  8  0 are
y
(1)
(2)
8
2, 4
(3)
(4)
3
1
x
2. Which of the following graphs illustrates a quadratic function that has no real zeros?
(1)
(3)
y
y
x
(2)
x
(4)
y
y
x
x
3. Determine the zeros for each quadratic function given below by using your graphing calculator. If
the function has no zeros, then so state.
(a) y  x 2  3x  10
(b) y   x 2  5 x  6
(c) y  x 2  8x  16
(d) y  x 2  2
(e) y   x 2  2 x  4
(f) y  25  10 x  x 2
Algebra 1, Unit #5 - Quadratic Functions – L4
The Arlington Algebra Project, LaGrangeville, NY 12540
4. A table of values is given below for the function y  27  6 x  x 2 . Use the table to determine the
values of x for which 27  6 x  x 2  0. Check by using the STORE command.
x
–10
–9
–8
–7
–6
–5
–4
–3
–2
–1
0
1
2
3
4
y
–13
0
11
20
27
32
35
36
35
32
27
20
11
0
–13
5. How many roots does x 2  7 x  6  0 have?
(1) 1
(2) 2
(3) 3
(4) 0
Reasoning
6. If the two x-intercepts of the graph of a quadratic function are 3 and 9, then the equation of the
axis of symmetry is
(1) x  6
(2) x  1
(3) x  3
(4) x  4
7. If one x-intercept of the graph of a quadratic function is 4 and the axis of symmetry has an equation
of x  7, then what is the other x-intercept?
8. The graph of a particular function of the form y  ax 2  bx  c, where a, b, and c are real numbers, is
shown below. Use the graph to solve each of the following equations.
y
x
(a) ax 2  bx  c  0
(b) ax 2  bx  c  1
Algebra 1, Unit #5 – Quadratic Functions – L4
The Arlington Algebra Project, LaGrangeville, NY 12540
(c) ax 2  bx  c  3