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**Equations Reducible to Quadratic**

Section 11.5 Equations Reducible to Quadratic Phong Chau

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Some equations are not quadratic, but can be turned into quadratic equation by using substitution. Such equations are called Equations in Quadratic Form or Reducible to Quadratic. x4 – 5x2 + 4 = 0 (x2)2 – 5(x2) + 4 = 0 u2 – 5u = 0.

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**Example Solve x4 – 5x2 + 4 = 0. Solution u2 – 5u + 4 = 0**

Let u = x2. Then we solve by substituting u for x2 and u2 for x4: u2 – 5u + 4 = 0 (u – 1)(u – 4) = 0 Factoring Principle of zero products u – 1 = 0 or u – 4 = 0 u = 1 or u = 4

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**Check: Replace u with x2 TRUE TRUE x2 = 1 or x2 = 4 x = 1: x = 2:**

(16) – 5(4) + 4 = 0 (1) – 5(1) + 4 = 0 TRUE TRUE The solutions are 1, –1, 2, and –2.

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**Example Solve Solution u2 – 8u – 9 = 0 (u – 9)(u +1) = 0**

Let u = Then we solve by substituting u for and u2 for x: u2 – 8u – 9 = 0 (u – 9)(u +1) = 0 u – 9 = 0 or u + 1 = 0 u = 9 or u = –1

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Check: x = 81: x = 1: FALSE TRUE The solution is 81.

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**Example Solve Solution u2 + 4u – 2 = 0**

Let u = t −1. Then we solve by substituting u for t −1 and u2 for t −2: u2 + 4u – 2 = 0

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Examples Solve the following equations:

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