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If the roots of the equation 24π₯ squared plus six π₯ plus π is equal to zero are not real, find the interval which contains π.
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So weβve been told that the roots of this quadratic equation, in which π is the constant term, are not real.
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We need to recall the relationship that exists between the coefficients of a quadratic equation and the type of roots that it has.
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Suppose we have the general quadratic equation ππ₯ squared plus ππ₯ plus π is equal to zero.
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The discriminant of a quadratic equation is the quantity π squared minus four ππ.
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The value or, more specifically, the sign of the discriminant is what determines the type of roots that the quadratic equation will have.
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If the discriminant is strictly positive, then the quadratic equation will have two real and distinct roots.
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If the value of the discriminant is equal to zero, then the quadratic equation has only one repeated real root.
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If the value of the discriminant is less than zero, then the quadratic equation has no real roots, which is the situation weβre given in this question.
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So we know then that the discriminant of this quadratic must be less than zero.
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Letβs work out what the discriminant is equal to in terms of π.
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Comparing the coefficients in our quadratic with the general form, we see that π is equal to 24, π is equal to six, and π is equal to π.
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Therefore, the discriminant π squared minus four ππ is equal to six squared minus four multiplied by 24 multiplied by π.
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This simplifies to 36 minus 96 π.
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Remember, the roots of this quadratic equation are not real.
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And so the value of the discriminant is less than zero.
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Therefore, we have the inequality 36 minus 96 π is less than zero.
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In order to find the interval which contains π, we need to solve this inequality for π.
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The first step is to subtract 36 from each side.
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This gives negative 96 π is less than negative 36.
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Next, we need to divide both sides of the inequality by negative 96.
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We need to be very careful here.
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Remember, when we divide an inequality by a negative number, we need to reverse the direction of the inequality.
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So the less than sign becomes a greater than sign.
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And we now have that π is greater than negative 36 over negative 96.
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The negative in the numerator and the negative in the denominator cancel out.
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And the fraction simplifies to three over eight, by dividing both the numerator and denominator by 12.
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We have then that π is greater than three over eight.
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The question doesnβt ask us to give our answer as an inequality.
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It asks us to give the interval which contains π.
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If π must be greater than three over eight, then the set of possible values of π is everything from three over eight to infinity.
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As the lower end of the interval is a strict inequality and the upper end is infinity, we can express this as an open interval, which is what the outward-facing square brackets indicate.
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π belongs in the open interval with end points three over eight and infinity.