USING THE FORMULA WHEN a IS NOT 1

USING THE FORMULA WHEN a IS NOT 1

What is it?

Keep a in 2a. For 2x² + 3x - 2 = 0, D = 9 + 16 = 25. x = -3 +/- 54. That gives x=12 or x=-2. Factoring (2x-1)(x+2) should match.

In simple words

• Keep a in 2a.
• DISCRIMINANT: D = b² - 4ac.
• TWO ROOTS: x = -3+54 = 24 = 12 and x = -3-54 = -84 = -2.
• Worked-method reminder: x = 24 = 12 or x = -84 = -2

Important terms

USING THE FORMULA WHEN a IS NOT 1: Keep a in 2a. For 2x^2 + 3x - 2 = 0, D = 9 + 16 = 25. x = -3 +/- 54. That gives x=12 or x=-2. Factoring (2x-1)(x+2) should match.

DISCRIMINANT: D = b^2 - 4ac. The 4ac includes a. For 2x^2+3x-2, ac is negative, so -4ac is positive and D is larger.

TWO ROOTS: x = -3+54 = 24 = 12 and x = -3-54 = -84 = -2. Check in the original: 2(14)+3(12)-2=12+32-2=0.

What you should learn

• Explain using the formula when a is not 1 using the correct definitions, conditions, and notation.
• Apply do not drop a, simplify +/- correctly in routine and exam-style questions.
• Show each important step, simplify the result, and check that it satisfies the original conditions.

Examples

• D=9-4(2)(-2)=25
• sqrt(25)=5
• Nice square

DISCRIMINANT

D = b^2 - 4ac. The 4ac includes a. For 2x^2+3x-2, ac is negative, so -4ac is positive and D is larger.

TWO ROOTS

x = -3+54 = 24 = 12 and x = -3-54 = -84 = -2. Check in the original: 2(14)+3(12)-2=12+32-2=0.

Full school-level explanation

Keep a in 2a. For 2x² + 3x - 2 = 0, D = 9 + 16 = 25. x = -3 +/- 54. That gives x=12 or x=-2. Factoring (2x-1)(x+2) should match.

Same formula, extra care with 2a:

WHAT — USING THE FORMULA WHEN a IS NOT 1

Keep a in 2a. For 2x² + 3x - 2 = 0, D = 9 + 16 = 25. x = -3 +/- 54. That gives x=12 or x=-2. Factoring (2x-1)(x+2) should match. The symbols and conditions are part of the mathematics, so include them when stating the rule.

WHY — Why this topic matters

Using The Formula When A Is Not 1 matters because quadratic equations model areas, motion, and many algebraic relationships. Knowing more than one solution method lets you choose the quickest reliable method and check the nature of the roots.

HOW — How to solve and explain it

Start by identifying what is given and what must be found. Work through DISCRIMINANT, then TWO ROOTS. Worked: solve 2x^2 + 3x - 2 = 0. a = 2, b = 3, c = -2 D = 3^2 - 4(2)(-2) D = 9 + 16 D = 25 x = -3 +/- 54 x = 24 = 12 or x = -84 = -2 Finish by checking signs, restrictions, units, and whether the answer matches the question.

DISCRIMINANT

D = b^2 - 4ac. The 4ac includes a. For 2x^2+3x-2, ac is negative, so -4ac is positive and D is larger.

Remember

• 4ac uses a

Examples

• D=9-4(2)(-2)=25
• sqrt(25)=5
• Nice square

TWO ROOTS

x = -3+54 = 24 = 12 and x = -3-54 = -84 = -2. Check in the original: 2(14)+3(12)-2=12+32-2=0.

Remember

• two substitutions

Examples

• x=12
• x=-2
• Check both

Remember

• do not drop a
• simplify +/-
• x = 12 or x = -2
• 2a not 2
• 4ac uses a
• two substitutions
• do not drop a
• simplify +/-

How the ideas connect

Worked: solve 2x² + 3x - 2 = 0. a = 2, b = 3, c = -2 D = 3² - 4(2)(-2) D = 9 + 16 D = 25 x = -3 +/- 54 x = 24 = 12 or x = -84 = -2

Deeper understanding

Go beyond memorising USING THE FORMULA WHEN a IS NOT 1. Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

Worked: solve 2x² + 3x - 2 = 0. a = 2, b = 3, c = -2 D = 3² - 4(2)(-2) D = 9 + 16 D = 25 x = -3 +/- 54 x = 24 = 12 or x = -84 = -2

Common mistakes

Exam focus

Core Idea

DISCRIMINANT

TWO ROOTS

Worked Method

Exam focus

Quick revision

• do not drop a
• simplify +/-
• x = 12 or x = -2
• 2a not 2
• USING THE FORMULA WHEN a IS NOT 1: Keep a in 2a. For 2x^2 + 3x - 2 = 0, D = 9 + 16 = 25. x = -3 +/- 54. That gives x=12 or x=-2. Fa
• DISCRIMINANT: D = b^2 - 4ac. The 4ac includes a. For 2x^2+3x-2, ac is negative, so -4ac is positive and D is large

Denominator is 2a = 4, not 2. Forgetting a is the usual slip.

For long answers on USING THE FORMULA WHEN a IS NOT 1: define clearly → explain the main parts → give one example → state one exam caution.