2.2.10

Quadratic Equations of Powers

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What are Quadratic Equations of Powers?

Quadratic equations of powers are equations of the form ap2+bp+cap^2 + bp+c where a,b,ca,b,c are constants and pp is a power of xx.

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What are quadratic equations of powers?

  • Quadratic equations of powers will always contain at least two terms:
    • A term with the highest power of xx: x2nx^{2n}.
    • The term with half the power of the highest power of xx: xnx^n.
  • Quadratic equations of powers can also include a constant term.
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How do you solve quadratic equations of powers?

  • In order to find the roots of a quadratic equation of a power, rewrite it so that it looks like a regular quadratic equation:
    • ax2n+bxn+c=0a(xn)2+b(xn)+c=0ax^{2n} + bx^n + c = 0\rightarrow a(x^n)^2 + b(x^n) +c = 0
  • We can then factorise this if it were a regular quadratic equation.
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Example

  • What is the solution to the equation f(x)=x616x3+64=0f(x) = x^6 - 16x^3 + 64= 0?
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Inspection

  • We can make this equation look more like a quadratic by grouping the factors of x3x^3 in the first two terms:
    • (x3)216(x3)+64=0(x^3)^2 - 16(x^3) + 64 = 0
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Factorise

  • We can factorise this equation as if it were a simple quadratic:
    • (x3)216(x3)+64(x38)2(x^3)^2 - 16(x^3) + 64 \equiv (x^3 - 8)^2
    • (x38)2=0(x^3 - 8)^2 = 0
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Solve for x3x^3

  • By taking the square root of both sides, we get:
    • x38=0x^3 - 8 = 0
  • Rearranging this we get:
    • x3=8x^3 = 8
  • Finally, we can take the cube root of both sides, which gives:
    • x=2x = 2

Jump to other topics

1Proof

2Algebra & Functions

2.1Powers & Roots

2.2Quadratic Equations

2.3Inequalities

2.4Polynomials

2.5Graphs

2.6Functions

2.7Transformation of Graphs

2.8Partial Fractions (A2 Only)

3Coordinate Geometry

4Sequences & Series

5Trigonometry

6Exponentials & Logarithms

7Differentiation

8Integration

9Numerical Methods

10Vectors

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