Quadratic Equations
Substitution of roots
Grade 11
Question:
<p>The value of the expression \(x^4 - 8x^3 + 18x^2 - 8x + 2\), when \(x = 2 + \sqrt{3}\), is</p>
<p>2</p>
<p>1</p>
<p>0</p>
<p>3</p>
Step-by-Step Solution
Key Concept: Recognize that x = 2 + √3 satisfies x² - 4x + 1 = 0, allowing you to reduce the quartic expression using this constraint rather than direct substitution.
<p><strong>Step 1:</strong> Verify that x = 2 + √3 satisfies a quadratic equation.</p><p>x - 2 = √3 ⟹ (x - 2)² = 3 ⟹ x² - 4x + 4 = 3 ⟹ <strong>x² - 4x + 1 = 0</strong></p><p><strong>Step 2:</strong> Express x⁴ and x³ using x² = 4x - 1.</p><p>x³ = x · x² = x(4x - 1) = 4x² - x = 4(4x - 1) - x = 16x - 4 - x = <strong>15x - 4</strong></p><p>x⁴ = x · x³ = x(15x - 4) = 15x² - 4x = 15(4x - 1) - 4x = 60x - 15 - 4x = <strong>56x - 15</strong></p><p><strong>Step 3:</strong> Substitute into the original expression.</p><p>x⁴ - 8x³ + 18x² - 8x + 2</p><p>= (56x - 15) - 8(15x - 4) + 18(4x - 1) - 8x + 2</p><p>= 56x - 15 - 120x + 32 + 72x - 18 - 8x + 2</p><p>= (56 - 120 + 72 - 8)x + (-15 + 32 - 18 + 2)</p><p>= 0·x + 1 = <strong>1</strong></p><p>∴ Answer: D</p>
Correct Answer: D