SAT-34: Determining Number and Type of Roots using the Discriminant

SAT-34: Determining Number and Type of Roots using the Discriminant

Description: The SAT often asks how many solutions an equation has, not what they are. The discriminant (b² − 4ac) answers this instantly, so you do not waste time solving.

The Three Cases

(Oʻzbekcha: diskriminantning ishorasi tenglama nechta haqiqiy yechimga ega ekanini koʻrsatadi.)

• b² − 4ac > 0 → two different real solutions (graph crosses the x-axis twice).
• b² − 4ac = 0 → exactly one real solution (graph just touches the x-axis).
• b² − 4ac < 0 → no real solutions (graph never touches the x-axis).

Memory hook: Positive = two, Zero = one, Negative = none. (Oʻzbekcha: musbat → ikkita, nol → bitta, manfiy → yechim yoʻq.)

Worked Example

How many real solutions does x² + 6x + 9 = 0 have?

• a = 1, b = 6, c = 9. Discriminant = 6² − 4(1)(9) = 36 − 36 = 0.
• Discriminant = 0 → exactly one real solution.

Harder SAT version (solve for a constant)

For what value of k does x² + kx + 16 = 0 have exactly one solution?

• One solution means discriminant = 0: k² − 4(1)(16) = 0.
• k² = 64 → k = ±8.

(Oʻzbekcha: "aniq bitta yechim" deganda diskriminantni 0 ga tenglab, nomaʼlumni topamiz.)

Practice

Does 2x² − 3x + 5 = 0 have real solutions?

Key words — Kalit soʻzlar

• Discriminant — diskriminant
• Real solution — haqiqiy yechim
• Root — ildiz
• x-axis — x oʻqi
• Graph — grafik
• Parabola — parabola
• Touch (the axis) — (oʻqqa) tegish
• Cross (the axis) — (oʻqni) kesib oʻtish
• Constant — oʻzgarmas (konstanta)
• Exactly one — aniq bitta

Summary

• Discriminant = b² − 4ac tells the number of real roots without solving.
• Positive → two, Zero → one, Negative → none.
• "Exactly one solution" → set the discriminant equal to 0 and solve for the unknown.