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Applications of Quadratics
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Using the discriminant Δ=b2−4ac to determine the number of real roots, solving quadratic inequalities from the roots and parabola shape, and applying quadratics to optimization problems.
The discriminant and solution count
SL AA 2.7
The discriminant of a quadratic is the term under the square root in the quadratic formula:
Δ=b2−4ac📖
When Δ<0, the square root has a negative value inside, and so the quadratic has no real solutions.
When Δ=0, the square root is zero, and the ±√Δ makes no difference, so there is only one real solution.
When Δ>0, √Δ is positive and so ±√Δ yields two real roots.
Quadratic Inequalities
SL AA 2.7
A quadratic inequality is an inequality of the form
ax2+bx+c{<≤>≥}0
They can be solved by finding the roots of the quadratic and the concavity of the parabola.
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