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Applications of Quadratics
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Using the discriminant Δ=b2−4ac to determine the number of real roots, Vieta's formulas α+β=−ab and αβ=ac, solving quadratic inequalities, and optimization problems modeled by quadratic functions.
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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.
Vieta's Formulas
AHL 2.12
In a quadratic ax2+bx+c, the roots α and β satisfy
α+β=−ab
αβ=ac
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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