SAT Math Problem (27)

Question

What is the sum of all values of m that satisfy 2m² - 16m + 8 = 0?

Answer 1

Special note:  This problem has a couple "tricks" that we often use to disguise an easy problem and make it look hard.  The goal is that students who know the material will answer the question correctly and students who don't know the material will miss the question.  If the question does not correctly discriminate between those two groups of students, it is not considered a good question --  right?

 

Trick 1:  Don't express the equation in simplest form.

   The problem says:     2m² - 16m + 8 = 0

   Simplest form:           m² - 8m + 4 = 0

 

Trick 2:  Use concepts that are not part of the standard ("the memorized") formula.

    Everyone knows (we hope) that for:

             For ax² + bx + c = 0

         ( -b ±√(b²-4ac) ) / 2a  are the two roots of a polynomial.

 

    Poor students won't know the sum of those two roots, but good students will "get it" somehow:

        ( -b +√(b2-4ac) ) / 2a    +    ( -b -√(b2-4ac) ) / 2a

        ( -b +√(b2-4ac)  -b -√(b2-4ac) ) / 2a

        -2b/2a

        -b/a

 

 For this problem:  -(-8)/1 = 8           [or you could have done:   -(-16)/2 = 8)

 

Standardized tests often have a "brick to stumble over" or a "brick wall to run into."  Since test makers think that using this technique makes good tests, there are lots and lots of "test taking tips" that clearly identify the "tricks" and thus have proved to increase individual scores by a very large amount [for both good and bad students].  That's why many teachers "teach to the test."

Answer 2

In general if x² + ax + b = 0, the sum of the values is -a.

 

For 2m² - 16m + 8 = 0 => m² - 8m + 4 = 0

The sum of the values is 8.

 

Check

m² - 8m + 4 = 0 has values m = (8 ± √(48)/2

                                              = (8 ± 4√(3))/2

                                              = 4 ± 2√(3)

 

Then the sum of the two values is (4 + 2√(13) + (4 -2 √(3)) = 8

Answer 3

 

    2m² - 16m + 8 = 0

 

Simplify by dividing both sides by 2

 

    m² - 8m + 4 = 0

   

Use complete the square to solve.  Take 1/2 of (-8) = -4, square it (-4)² = 16, and add to both sides

 

    m² -8m = -4

    m² -8m + 16 = -4 + 16

    (m - 4)(m - 4) = 12

    (m - 4)² = 12

 

Take the square root of both sides

 

   √ (m - 4)² = ±√12

    m - 4 = ±(√4 •√3)

    m - 4 = ±2√3

    m = 4 ± 2√3

 

The sum of the solutions are:

 

     4 + 2√3 + 4 -  2√3

  = 8

 

Answer 4

I notice that the coefficients and constants are all even and therefore divisible by 2.

Attempt 1: Solving the quadratic by Factoring

2m² – 16m + 8 = 0

I factor out 2 from each term on the RHS:

2(m² - 8m + 4) = 0

Divide both sides of the equation by 2:

m² - 8m + 4 = 0

When a quadratic is of the form ax² + bx + c where a, b, and c are constants, we look for a pair of numbers r and s such that their sum r + s = b and their product rs = ac.

In this case,

a = 1

b = -8

c = 4

Therefore, let’s consider the factor pairs of 4: 1 & 4, 2 & 2. Does either factor pair add up to -8? No. The sum of two positive numbers must be positive, not negative. Even if we considered -1 & -4; & -2 & -2, neither of these pairs add up to -8.

Attempt 2: Quadratic Formula

x = (-b ± √(b^2 - 4ac)) / 2a

x = (8 ± √(64 -4(4)) / 2

x = (8 ± √(48) / 2

x = (8 ± √(16*3) / 2

x = (8 ± √(16)*√(3)) / 2

x = (8 ± 4 * √(3)) / 2

x = 4 ± 2 √(3)

Sum of solutions:

4 + 2 √(3) + 4 - 2 √(3) =

8

The sum of all values of m that satisfy 2m² - 16m + 8 = 0 is 8.

Answer 5

Pari,

 

Several ways to do this; easiest is to use the quadratic formula to solve for m.  You'll find two solutions to m; just add them together.