Question

please state the reasons for the ans..
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Answer 1

What is a quadratic equation?

• A quadratic equation is a polynomial whose standard form is ax² + bx + c where a, b and c are real numbers and a ≠ 0.

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(2.) Which of the following is a quadratic equation?

• (x² + 1) = (2 - x)² + 3

$\implies$ x² + 1 = 4 + x² - 4x + 3

$\implies$ x² - x² + 4x = 4 + 3 - 1

$\implies$ 4x = 6

Therefore, it is not a quadratic equation.

• x³ - x² = (x - 1)³

$\implies$ x³ - x² = x³ - 3x² + 3x - 1

$\implies$ x³ - x³ - x² + 3x² - 3x + 1 = 0

$\implies$ 2x² - 3x + 1

Therefore, it is a quadratic equation.

• 2x² + 3 = (5 + x) (2x - 3)

$\implies$ 2x² + 3 = 10x - 15 + 2x² - 3x

$\implies$ 2x² - 2x² - 10x + 3x + 3 + 15

$\implies$ -7x + 18

Therefore, it is not a quadratic equation.

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(3.) Which of the following is not a quadratic equation?

• 3x - x² = x² + 5

$\implies$ 3x - x² - x² - 5 = 0

$\implies$ -2x² + 3x - 5

Therefore, it is a quadratic equation.

• (x + 2)² = 2 (x² - 5)

$\implies$ x² + 4 + 4x = 2x² - 10

$\implies$ x² - 2x² + 4x + 4 + 10

$\implies$ -x² + 4x + 14

Therefore, it is a quadratic equation.

• (√2x + 3)² = 2x² + 6

$\implies$ 2x² + 9 + 2√2x = 2x² + 6

$\implies$ 2x² - 2x² + 9 - 6 + 2√2x

$\implies$ 3 + 2√2x

Therefore, it is not a quadratic equation.

• (x - 1)² = 3x² + x - 2

$\implies$ x² - 2x + 1 = 3x² + x - 2

$\implies$ x² - 3x² - 2x - x + 1 + 2

$\implies$ -2x² - 3x + 3

Therefore, it is a quadratic equation.

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The answers are :

• (2.) (b)
• (3.) (c)