Answer:
x = -2/5 or 2
Explanation:
You want to factor and solve the quadratic equation 5x² -8x -4 = 0.
Factors
When the leading coefficient is not 1, the factoring process is a little different than when the leading coefficient is 1.
Form the product of the leading coefficient and the constant:
(5)(-4) = -20
Find the divisor pairs of this product. You want their sum to have the value of coefficient of the middle term (-8).
-20 = -20(1) = -10(2) = -5(4) . . . . . . . sums are -19, -8, -1
The pair of factors that have a sum equal to the x-term's coefficient is the pair you are interested in. At this point, you can use this information two ways.
1) Factor by grouping
Rewriting the middle term using these values you have ...
5x² -10x +2x -4 = 0
Factoring pairs of terms gives ...
(5x² -10x) +(2x -4) = 0
5x(x -2) +2(x -2) = 0 . . . . find the common factor of each pair of terms
(5x +2)(x -2) = 0 . . . . . . . factor out the common factor
2) Copy the leading coefficient, then remove it
Writing binomials that use the factors of -20 found above we have ...
(5x -10)(5x +2) . . . . . . . note the leading coefficient is repeated in each binomial
We want to remove this extra factor of 5, so our equation is actuallly ...
(5x -10)(5x +2)/5 = 0
We note that there is an unneeded factor of 5 in the first binomial (5x-10), so we can simplify this to ...
(x -2)(5x +2) = 0
This is the same factoring of the quadratic that we found using the first method.
Zero product rule
When you have the factored form of the equation telling you the product of factors is zero, the "zero product rule" applies. That rule says the product will only be zero if a factor is zero. This gives the two solutions:
x -2 = 0 ⇒ x = 2
5x +2 = 0 ⇒ x = -2/5
The solutions to the quadratic are x = -2/5 or x = 2.
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Additional comment
For quadratic ax² +bx +c, the sum of the solutions is -b/a, and the product of the solutions is c/a. Knowing this means you can choose the correct answer without actually working the problem.
The products of the offered answer choices are (-5/2)(-2) = 5; (-2/5)(-2) = 4/5; (-2/5)(2) = -4/5; (2/5)(2) = 4/5. The value of c/a is -4/5, so all answer choices are eliminated except the correct one: -2/5 or 2.
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