SAT-33: The Quadratic Formula and the Discriminant

SAT-33: The Quadratic Formula and the Discriminant

Description: Some quadratics do not factor nicely. The quadratic formula solves every quadratic, even the ugly ones. It is one of the most important tools on the SAT.

The Formula

For ax² + bx + c = 0:
x = ( −b ± √(b² − 4ac) ) / (2a)

The ± means you usually get two answers: one with + and one with −. (Oʻzbekcha: ± belgisi odatda ikkita javob beradi.)

The Discriminant

The part under the square root, b² − 4ac, is called the discriminant. You will use it constantly in SAT-34.

(Oʻzbekcha: ildiz ostidagi qism — b² − 4ac — diskriminant deyiladi.)

Worked Example

Solve 2x² + 3x − 5 = 0. Here a = 2, b = 3, c = −5.

(Oʻzbekcha: avval a, b, c larni ishoralari bilan toʻgʻri yozib oling, keyin formulaga qoʻying.)

• Discriminant: b² − 4ac = 3² − 4(2)(−5) = 9 + 40 = 49.
• x = ( −3 ± √49 ) / (2·2) = ( −3 ± 7 ) / 4
• x = 4/4 = 1 or x = −10/4 = −5/2

Tip: write down a, b, c carefully with their signs before plugging in. A wrong sign is the most common mistake here.

Practice

Solve x² − 4x + 1 = 0 (it does not factor).

Key words — Kalit soʻzlar

• Quadratic formula — kvadrat tenglama formulasi
• Discriminant — diskriminant
• Root / Solution — ildiz / yechim
• Square root — kvadrat ildiz
• Coefficient — koeffitsiyent
• Plus-minus (±) — qoʻshish-ayirish belgisi
• Substitute / Plug in — (formulaga) qoʻyish
• Real number — haqiqiy son
• Factor — koʻpaytuvchilarga ajratish

Summary

• Quadratic formula: x = (−b ± √(b² − 4ac)) / (2a) — works for any quadratic.
• The discriminant is b² − 4ac (under the root).
• Identify a, b, c with correct signs first.
• The ± normally gives two solutions.