How do you find the roots, real and imaginary, of #y=(x/5-1)(-2x+5)# using the quadratic formula?
No need to use the quadratic formula to find the roots since the equation is already factored out. Equate y to 0 to find the roots.
If you really need to use the quadratic formula
Given the quadratic equation
Proceeding with the computation should give us the same result as with the one above.
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To find the roots of the given quadratic equation ( y = \left(\frac{x}{5} - 1\right)(-2x + 5) ) using the quadratic formula:
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Rewrite the equation in the standard form ( ax^2 + bx + c = 0 ): ( y = -\frac{2}{5}x^2 + \frac{12}{5}x - 5 )
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Identify the coefficients: ( a = -\frac{2}{5} ), ( b = \frac{12}{5} ), ( c = -5 ).
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Apply the quadratic formula: ( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} ).
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Substitute the coefficients into the formula: ( x = \frac{{-\frac{12}{5} \pm \sqrt{{\left(\frac{12}{5}\right)^2 - 4 \cdot \left(-\frac{2}{5}\right) \cdot (-5)}}}}{{2 \cdot \left(-\frac{2}{5}\right)}} )
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Simplify: ( x = \frac{{-\frac{12}{5} \pm \sqrt{{\frac{144}{25} - \frac{40}{5}}}}}{{-\frac{4}{5}}} ) ( x = \frac{{-\frac{12}{5} \pm \sqrt{{\frac{144}{25} - \frac{200}{25}}}}}{{-\frac{4}{5}}} ) ( x = \frac{{-\frac{12}{5} \pm \sqrt{{-\frac{56}{25}}}}}{{-\frac{4}{5}}} )
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Simplify further: ( x = \frac{{-\frac{12}{5} \pm \frac{{\sqrt{56}}}{5}i}}{{-\frac{4}{5}}} ) ( x = \frac{{-12 \pm \sqrt{56}i}}{{-4}} ) ( x = \frac{3}{2} \pm \frac{{\sqrt{14}i}}{2} )
Therefore, the roots of the equation ( y = \left(\frac{x}{5} - 1\right)(-2x + 5) ) are ( x = \frac{3}{2} + \frac{{\sqrt{14}i}}{2} ) and ( x = \frac{3}{2} - \frac{{\sqrt{14}i}}{2} ).
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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