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Quadratic equations

3. Solving Quadratic Equations by Completing the Square—ANY quadratic
equation can be solved by completing the square. The method works
when the factors are not immediately obvious, making the equation
difficult to factor.
Step 1: Keep the x2 and x terms on one side of the equation, and the
constant term on the other.
Step 2: Complete the square. If we have Ax2 + Bx = -C
complete its square by adding the constant A(B/2A)2 to both sides.
In other words, we complete the square by taking ½ of B/A , squaring it and
further scaling it by A. Be sure to add it to BOTH sides to keep the equation
balanced.

Step 3: Factor the perfect square (created in step 2) as a binomial
squared :
or dividing both sides by A:
Step 4: Now we can solve the equation in step 3 by using the square root
method outlined previously.
Example 8: Solve x2 - 8x = -11 by completing the square.
Step 1: Keep the x2 and x terms on one side of the equation, and the
constant term on the other. Already done.
Step 2: Complete the square.
Step 3: Factor the perfect square (created in step 2) as a binomial
squared :
Step 4: Now we can solve the equation in step 3 by using the square root
method outlined previously.
Example 9: Solve 3x2 + 8x – 12 = 0 by completing the square.
Step 1: Keep the x2 and x terms on one side of the equation, and the
constant term on the other.
Step 2: Complete the square. For 3x2 + 5x = 12, A = 3 and B = 5. Thus,
( also ) so, to keep the equation
balanced,
Step 3: Step 3: Factor the perfect square (created in step 2) as a binomial
squared :
Step 4: Now we can solve the equation in step 3 by using the square root
method outlined previously.
4 Solving Quadratic Equations by the Quadratic Formula ANY quadratic equation
can be solved by using the quadratic formula. If the factors do not
immediately present themselves, this formula will nail them.
Step 1: Simplify each side if needed. To remove fractions: As fractions
are another way to perform division, and the inverse of division is
multiplication, you remove fractions by multiplying both sides by the
least common denominator of all of your fractions.
Step 2: Write in standard form, Ax2 + Bx + C , if needed. If it is not in
standard form, move any term(s) to the appropriate side by using
the addition/subtraction property of equality.
Step 3: Identify A, B, and C. When the quadratic equation is in standard
form, then A is the coefficient in front of the x2 term, B is the
coefficient in front of the x term, and C is the constant term.
Step 4: Use A, B, and C in the quadratic formula. Roots
Step 5: Simplify if possible.
Example 10: Solve 4x2 – 6 = 3x by using the quadratic formula.
Step 1: Simplify each side if needed. No fractions, already simple
Step 2: Write in standard form, if needed.
Step 3: Identify A, B, and C. A = 4 B = -3 C = -6
Step 4: Use A, B, and C in the quadratic formula.
Step 5: Simplify if possible.
Example 11: Solve x(x + 2) = -10 by using the quadratic formula.
Step 1: Simplify each side, if needed.
Step 2: Write in standard form, if needed.
Step 3: Identify A, B, and C. A = 1 B = 2 C = 10
Step 4: Use A, B, and C in the quadratic formula.
Roots x =
Step 5: Simplify if possible.
Example 12: Solve by using the quadratic formula.
Step 1: Simplify each side, if needed.
Step 2: Write in standard form, if needed.
Step 3: Identify A, B, and C.
Step 4: Use A, B, and C in the quadratic formula.

Roots x =
Step 5: Simplify if possible. Multiply numerator and denominator by 2:

Discriminant—When a quadratic equation is in standard form, Ax2 + Bx + C,
= 0 the expression, B2- 4AC, that is found under the square root part of the
quadratic formula is called the discriminant. The sign of the discriminant
can reveal how many solutions there are going to be and if the solutions are
real numbers or complex numbers.

When the Discriminant
b2-4ac is
Characteristic
Solution
Comment
> 0 Two distinct
real solutions
As the discriminant is found
under the square root, if its
value is positive, then the
quadratic possesses two distinct
real roots
= 0 One real
solution
As the discriminant is found
under the square root, if its
value is zero, then the quadratic
possesses only one real root.
< 0 Two distinct
complex
imaginary
solutions
As the discriminant is found
under the square root, if its
value is positive, then the
quadratic possesses two distinct
complex roots

To determine the determinant, it is necessary to convert a given trinomial
form to standard form. Then you can easily read off A, B and C.

For the following quadratics, determine the value of the discriminant and
give the number and type of roots:

(a) 2x2 – 8x + 7 discr. = 82 - 4(2)(7) = 64 - 56 = 8, >0,

(b) x2 + x - 1 discr. =

(c) x2 + 4x + 4 discr. =

(d)   Determine over the entire domain 0 ≤ θ ≤ ½π

discrim = [2cos(θ)]2 – 4( ½)sin(2θ)

=

Now from a trigonometric identity:

Note that the discriminant is a function of θ. For some values of θ, it is
positive, for others it is negative. So, we must divide the domain into 3
regions:

Now for 0 < θ < 45°, cos(θ) > sin(θ), whereas for 45° < θ <90, cos(θ) < sin (θ).

Thus for 0 < θ < 45°.

When θ = 45°, cos(45°) = sin(45 °), thus

For 45° < θ < 90° cos(θ) < sin(θ) so

(e) 3x2 – 5x + 2

(f) x2 + 4 = 0 discr. =

(g) An alternate form of the quadratic equation is given by dividing all the terms
the standard form of the quadratic equation by x2 to obtain;

This form is helpful if B2 >> 4AC where denotes much greater, in which case
the usual form of the quadratic formula can give inaccurate numerical results
for one of the roots. Show that it leads to the solution:

Solution. Let’s solve by completing the square:

Step 1: Keep the x -2 and x -1 terms on one side of the equation, and the
constant term on the other.

Step 2: Complete the square=

Inside the parentheses, take half of and square it, add result to both sides:

Step 3: Factor the perfect square (created in step 2) as a binomial squared:

Divide by C :

Step 4: Now we can solve the equation in step 3 by using the square root
method outlined previously.

   subtract from both sides:

Finally, taking the inverse to recover x :

x =

Okay-eee, that should just about do it—Either you love quadratics more or
you loathe them more now.