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John completes a job in x hours, while Rachel completes the same job in (x+4) hours. If they work together, they can complete the job in 3 hours and 45 minutes.

By forming an equation in x, show that 2x^2-7x-30=0.

John: x hours to complete job means he works at a rate of 1/x.

Rachel: x+4 hours to complete the job means he works at a rate of 1/(x+4).

For both working together, they work at a rate of 1/x + 1/(x+4), and that take 3.75 hours.

This says we have 1/[1/x + 1/(x+4)] = 3.75, and 3.75 = 15/4, so we have.

1/[1/x + 1/(x+4)] = 15/4.

Inverting both side gives 1/x + 1/(x+4) = 4/15.

Multiplying both sides by 15x(x+4) gives 15(x+4) + 15x = 4x(x+4).

Multiplying out each side of the equation gives 15x + 60 + 15x = 4x² + 16x.

Adding 15x + 15x on the left side gives 30x + 60 = 4x² + 16x.

Adding -30x - 60 to both sides gives 0 = 4x² - 14x - 60.

Dividing each term by 2 gives 0 = 2x² - 7x - 30.

That is what we are given to show.

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