Decide equation using only factors to solve this problem 12x^{2} + 5x - 2 = 0

Question
Comparing two groups
asked 2021-03-09
Decide equation using only factors to solve this problem
\(12x^{2}\ +\ 5x\ -\ 2 = 0\)

Answers (1)

2021-03-10
In order to factor the quadratic, we need to find the values of a, b and c for the given quadratic \(12x^{2}\ +\ 5x\ -\ 2 = 0\) on comparing with standard quadratic
\(ax^{2}\ +\ bx\ +\ c = 0\)
\(12x^{2}\ +\ 5x\ -\ 2 = 0\)
\(a = 12\, b = 5,\ c =\ -2.\)
Now, let us apply ac method to factor the quadratic.
We got \(ac=\ -24\ and\ b=5.\) So, we need to find the factors of -24, those add upto 5.
\(ac = 12 \cdot\ -2 =\ -24\)(Product)
\(b = 5\)(Sum)
-24 could be factored in two numbers 8 and -3 that add upto 5.
Therefore, middle term 5x of expression \(12x^{2}\ +\ 5x\ -\ 2 =0,\) could be break into two terms 8x and -3x.
Then we need to make the break the expression into groups.
Factor out 3x form \(12x^{2}\ -\ 3x\) and factor out 2 from:
\(8x\ -\ 2.\)
Then factor out common parenthesis \((4x\ -\ 1).\)
We got factored form \((4x\ -\ 1) (3x\ +\ 2) = 0\)
\(12x^{2}\ +\ 5x\ -\ 2 = 0\)
\((12x^{2}\ -\ 3x)\ +\ (8x\ -\ 2) = 0\)
\(3x(4x\ -\ 1)\ +\ 2 (4x\ -\ 1) = 0\)
Factor out common parenthesis \((4x\ -\ 1)\)
\((4x\ -\ 1)(3x\ +\ 2) = 0\)
By applying zero product rule, we need to set each factor equal to zero and solve for x.
We got \(x = \frac{1}{4},\ \frac{-2}{3}.\)
\(4x\ -\ 1 = 0\ or\ 3x\ +\ 2 = 0\)
\(+\ 1\ +\ 1\ -2\ -2\\)
\(\frac{4x}{4} = \frac{1}{4}\ or\ \frac{3x}{3} = \frac{-2}{3}\)
\(x = 1.4\ or\ x = \frac{-2}{3}\)
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Relevant Questions

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The table below shows the number of people for three different race groups who were shot by police that were either armed or unarmed. These values are very close to the exact numbers. They have been changed slightly for each student to get a unique problem.
Suspect was Armed:
Black - 543
White - 1176
Hispanic - 378
Total - 2097
Suspect was unarmed:
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White - 67
Hispanic - 38
Total - 165
Total:
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Think of it this way. If you went to a college that was 90% female and 10% male, then females would most likely have the highest percentage of A grades. They would also most likely have the highest percentage of B, C, D and F grades
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