Answer: The square root of 169 is 13, so the two solutions for x are:
x = (13 + 13) / 4 = 3.25
x = (13 - 13) / 4 = -0.25
Since the quadratic expression 2x^2 - 13x - 5 equals 0 when x is either 3.25 or -0.25, the factors of this expression are (x - 3.25)(x + 0.25).
In summary, the factors of the first quadratic expression are (x + 1.056)(x + 5.937), the factors of the second expression are (x - 0.75)(x + 3.75), and the factors of the third expression are (x - 3.25)(x + 0.25).
Step-by-step explanation: To find the factors of a quadratic expression, we can use the quadratic formula. The quadratic formula states that for a quadratic expression of the form ax^2 + bx + c = 0, the solutions for x are given by the formula x = (-b ± sqrt(b^2 - 4ac)) / (2a).
In the first equation, a is 9, b is 31, and c is 20. Plugging these values into the quadratic formula, we get:
x = (-31 ± sqrt(31^2 - 4 * 9 * 20)) / (2 * 9)
This simplifies to:
x = (-31 ± sqrt(961 - 720)) / 18
This further simplifies to:
x = (-31 ± sqrt(241)) / 18
We can find the value of the square root by using a calculator or by remembering some common square roots. The square root of 241 is approximately 15.937, so the two solutions for x are:
x = (-31 + 15.937) / 18 = -1.056
x = (-31 - 15.937) / 18 = -5.937
Since the quadratic expression 9x^2 + 31x + 20 equals 0 when x is either -1.056 or -5.937, the factors of this expression are (x + 1.056)(x + 5.937).
We can use the same approach to find the factors of the other two quadratic expressions. For the second expression 2x^2 + 7x - 4, the values of a, b, and c are 2, 7, and -4, respectively. Plugging these values into the quadratic formula, we get:
x = (-7 ± sqrt(7^2 - 4 * 2 * -4)) / (2 * 2)
This simplifies to:
x = (-7 ± sqrt(49 + 32)) / 4
This further simplifies to:
x = (-7 ± sqrt(81)) / 4
The square root of 81 is 9, so the two solutions for x are:
x = (-7 + 9) / 4 = 0.75
x = (-7 - 9) / 4 = -3.75
Since the quadratic expression 2x^2 + 7x - 4 equals 0 when x is either 0.75 or -3.75, the factors of this expression are (x - 0.75)(x + 3.75).
For the third expression 2x^2 - 13x - 5, the values of a, b, and c are 2, -13, and -5, respectively. Plugging these values into the quadratic formula, we get:
x = (13 ± sqrt(13^2 - 4 * 2 * -5)) / (2 * 2)
This simplifies to:
x = (13 ± sqrt(169 + 40)) / 4
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