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Solve π₯ squared minus three π₯ minus four is less than or equal to zero.
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This is a quadratic inequality.
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And if we think about quadratic inequalities, we know that they have certain features.
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For the general form ππ₯ squared plus ππ₯ plus π, when π is positive, the parabola opens upward.
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We also know that the points where the parabola crosses the π₯-axis are called the roots.
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When weβre thinking about quadratic inequalities, the behavior around the roots is what we want to consider since the behavior of the graph changes on either side of the roots.
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This means to solve this problem, our first step will be to try and identify the roots.
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We wanna let the equation be equal to zero, as thatβs where the roots will be.
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If we solve by factoring, weβre looking for values that multiply together to equal negative four and add together to equal negative three.
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That would be negative four and positive one.
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We set both of those factors equal to zero, and we find that π₯ is equal to negative one and that π₯ is equal to positive four.
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If we think about our π₯-axis with roots at negative one and positive four, we want to check the behavior on either side of the roots and between them.
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We know that zero falls between negative one and four.
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So, we plug in zero to our original equation.
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We want to know is zero squared minus three times zero minus four less than or equal to zero?
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That is negative four.
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And negative four is less than or equal to zero.
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Based on this, we can say that the function between negative one and four will be negative.
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We now want to check either side of the roots.
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We can check positive five.
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Five squared minus three times five minus four is less than or equal to zero.
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That gives us six.
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We know that six is not less than or equal to zero.
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π₯ squared minus three π₯ minus four will be positive for all π₯-values greater than four.
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Weβll check to the left of negative one at negative two, which again gives us positive six.
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Positive six is not less than zero.
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π₯ squared minus three π₯ minus four will be positive for all values less than negative one.
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We are interested in the places where this function is less than or equal to zero, the places where itβs negative or equal to zero.
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And in this case, it will fall between the roots negative one and four.
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We use this type of inclusive brackets because negative one and four are also part of the solution.
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And we can say that this quadratic inequality is true for negative one to four.