Download SM2H Unit 3 – Solving Quadratic Equations

SM2H Unit 3 – Solving Quadratic Equations
3.1 Introduction to Quadratics
A polynomial (in one variable) that has a degree of 2, (the largest exponent is a 2) is called a
Quadratic. The following are the different forms of a Quadratic Equation.
Standard Form:
ax2  bx  c  0, where a  0
ex: 4 x 2  5 x  3  0
Vertex Form:
What is b?
What is c?
What is h?
What is k?
What is p?
What is q?
a( x  h)2  k  0, where a  0
ex: 0  3( x  4)2  7
Factored Form:
What is a?
What is a?
a( x  p)( x  q)  0, where a  0
ex: ( x  6)( x  2)  0
What is a?
Why have three different forms of a Quadratic Equation?
Each form of a quadratic equation gives us easy access to important information – either about
the solutions of the equation or about the graph of the quadratic function.
The graph of a quadratic function is called a parabola.
(You’ll learn more about this later.) The parabola is a
symmetrical “u” shape.
The Vertex Form – makes it easy to find the vertex – the
maximum or minimum point of the graph. The
coordinates of the vertex are  h, k  .
From the Standard Form it is easy to find the y-intercept
of the graph, or the place where the parabola crosses the yaxis. The coordinates of the y-intercept are  0,c 
From the Factored Form of a quadratic equation – it is easy to find x-intercepts of the graph, or
the point(s) where the parabola crosses the x-axis. The coordinates of the x-intercepts are  p,0
and  q, 0  . The x-intercepts are also called solutions, zeros or roots.
In this unit, we will focus on finding the solutions (or zeros or roots) of a quadratic equation.
3.2 The Quadratic Formula and the Discriminant
The Quadratic Formula: A quadratic equation written in the form ax 2  bx  c  0 , where a  0, has
the solutions:
b  b2  4ac
x
.
2a
Solving a Quadratic Equation Using the Quadratic Formula:
1.
2.
3.
4.
Write the equation in standard form: ax 2  bx  c  0.
Identify a, b, and c. Plug them into the equation. Be careful with parentheses.
Simplify. Be careful to follow order of operations and deal with negatives correctly.
Write your answers in simplified radical form. Use i when appropriate.
Examples: Solve each equation using the quadratic formula.
a) x 2  4 x  7  0
b) 3m 2  16m  5  0
c) 2 w2  4 w  3
d) n 2  4n  4  0
e) r 2  9  0
f) 6u 2  2u  0
g) z  3z 2  3
h)
1 2
1
y  y 0
4
2
Discriminant: The radicand of the quadratic equation, b 2  4ac .
The discriminant tells us about the number and types of solutions of a quadratic equation without
actually solving it. It also tells us how many x-intercepts the graph of a function has.
Discriminant: b  4ac
Solutions of ax  bx  c  0
Positive
Two real solutions
Zero
One real solution
Negative
Two imaginary solutions
2
2
Examples: Find the discriminant of each quadratic equation and state the number and type (real or
imaginary) of solutions.
a) 4 x 2  20 x  25  0
b) x 2  2 x  4  0
c) 3x 2  5  7 x
d) x 2  5 x  14
3.3 The Square Root Principle
Example: How many numbers can be squared to get 9? In other words, how many solutions are there to
the equation x 2  9 ? What are they? What about the equation x 2  9?
All numbers except zero have two square roots, a positive square root and a negative square root.
The
symbol means the positive square root. Both roots must be considered when solving an
equation by taking square roots, so we use the  symbol to include both roots.




The number i: i is the number whose square is –1. That is, i  1 and i 2  1.
Imaginary Number: A number that can be written in the form a  bi, where a and b are
real numbers and b  0. Any number with an i in it is imaginary.
Square Root Property: If b is a real number and if a 2  b, then a   b.
Radicand: The number under the radical sign.
Solving Equations by Taking Square Roots: Do this when the equation has a perfect square and no other
variables.
1.
2.
3.
4.
Get the perfect square alone on one side of the equation.
Use the square root property.
Simplify all square roots. Write the square roots of negative numbers in terms of i.
Solve for the variable, if necessary.
Examples: Solve each equation using the square root property. Include both real and imaginary
solutions. Write your solutions in simplest radical form. Write imaginary solutions in the form a  bi.
a) x 2  50
b) 3d 2  48
c) 2 z 2  48
d) p 2  3  42
e) 2n 2  4  104
f) 3q 2  6  48
g) 16   y  1
h)  2m  5   25
i)
 r  4
2
2
 10  26
2
j) 10 
1
2
n  7
2
k) 5  x  10   0
l) 4  w  3  6  86
m) 2  x  3  32
n) 16  
2
2
2
1
2
 x  2
3
Flashback -- Multiplying Trinomials!!!
A Perfect Square Trinomial comes from multiplying two identical binomials together.
Ex:
 r  4
2
  r  4  r  4   r 2  8r  16
So if we had  r  4   25 we could easily solve it by taking the Square Root of each side of the
2
equation.
What if the trinomial that we have in an equation we are trying to solve is NOT a Perfect Square
Trinomial? Can we turn it into one so that we can take the Square Root of each side to solve it?
3.4 Completing the Square
To Solve a Quadratic Equation in x by Completing the Square:
1. Isolate the terms with variables on one side of the equation, and arrange them in descending
order.
2. Divide both sides of the equation by the coefficient of x 2 , if that coefficient is not 1.
3. Divide the coefficient of x by 2 then square your answer. Add the result to both sides of the
equation. This is called completing the square.
To complete the square for x 2  bx, add
 b2 
2
.
4. Factor the resulting perfect square trinomial and write it as the square of a binomial.
x 2  bx   b2    x  b2 
2
2
5. Use the principle of square roots and solve for x.
Examples: Replace the blanks in each equation with constants to form a true equation.
2
a) x  18 x  ___   x  ___ 
2
b) x  3x  ___   x  ___ 
2
2
c) x 2  53 x  ___   x  ___ 
2
Examples: Solve by completing the square. Show your work.
2
a) x  4 x  12  0
2
b) x  6 x  6  0
2
c) 4 x  8 x  3  0
2
d) x  16 x  59  7
3.5 Mid-Unit Review
How did the Quadratic Formula come about? Well, it turns out that it can be derived by taking the
Standard Form of a Quadratic Equation and then solve by Completing the Square.
How do I Select the Appropriate Method to Use?
Vertex form – solve by using the Square Root Principle (i.e. take the square root of both sides of the
equation.)
Standard form – Use the Quadratic Formula or Complete the Square