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Section A.3 Quadratic equations

We will be solving quadratic equations as we solve differential equations. If we want to solve a quadratic equation like ax2+bx+c=0, there are several different methods we might use, including:
  1. factoring
  2. quadratic formula, x=b±b24ac2a
  3. completing the square
Most students prefer the first two methods, which is fine. We will end up completing the square later in the semester, so if you want to review that method now, you’ll reap the benefits later!
Solve the following quadratic equations. Note: It’s OK if the solutions are complex or imaginary.
  1. 4x2+12x+9=0
    Solution. Solution
    You might solve via factoring:
    4x2+12x+9= 0(2x+3)(2x+3)= 02x+3=0or2x+3=0x=32,orx=32x=32 (double root)
    Alternately, you might use the quadratic formula:
    x= b±b24ac2a= 12±1224(4)(9)2(4)= 12±1441448= 12±08(0 implies a double root)= 128= 32
    You could even complete the square:
    4x2+12x+9= 04x2+12x= 9x2+3x= 94x2+3x+94= 94+94(x+32)2= 0x+32= ±0x= 32 (double root) 
    Answer. Answer
    x=32 (double root)
  2. 2x29x35=0
    Solution. Solution
    You might solve via factoring:
    2x29x35= 0(2x+5)(x7)= 02x+5=0orx7=0x=52,orx=7x=52,7
    Alternately, you might use the quadratic formula:
    x= b±b24ac2a= (9)±(9)24(2)(35)2(2)= 9±81+2804= 9±3614= 9±194= 9+194,9194= 284,104= 7,52
    You could even complete the square:
    2x29x35= 02x29x= 35x292x= 352x292x+(94)2= 352+(94)2(x94)2= 28016+8116(x94)2= 36116x94= ±36116x94= ±194x= 94±194x= 94+194,94194x= 284,104x= 7,52
    Answer. Answer
    x=52,7
  3. x24x+13=0
    Solution. Solution
    This one doesn’t factor easily... You might use the quadratic formula:
    x= b±b24ac2a= (4)±(4)24(1)(13)2(1)= 4±16522= 4±362= 4±6i2= 2±3i= 2+3i,23i
    You could even complete the square:
    x24x+13= 0x24x= 13x24x+4= 13+4(x2)2= 9x2= ±9x2= ±3ix= 2±3i= 2+3i,23i
    Answer. Answer
    x=2±3i
  4. Name at least two methods for solving quadratic equations.
    Answer. Answer
    factoring, using the quadratic formula, completing the square
  5. How many solutions does a quadratic equation have?
    Answer. Answer
    There are three possible outcomes when solving a quadratic equation:
    1. two distinct real roots
    2. one repeated real root (i.e., a double root)
    3. complex conjugate roots

Solving Quadratic Equations.

The solution to the quadratic equation
(52)ax2+bx+c=0
is given by the quadratic formula:
(53)x=b±b24ac2a.
Notes:
  1. The ± gives two solutions, say x1 and x2.
  2. x1 and x2 are also known as the roots of  ax2+bx+c.
  3. The value, b24ac , under the root in is called the discriminant.
  4. Equation (52) can be written as (xx1)(xx2)=0.
  5. If b24ac>0, then x1 and x2 are different real numbers.
  6. If b24ac=0, then x1 and x2 are the same real number (repeated).
  7. If b24ac<0, then x1 and x2 are complex and can be written as
    x1=α+βi,x2=αβi.
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