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Rationalize Denominators & Solve Quadratic Equations: Examples & Practice

Learn to simplify fractions & solve quadratic equations with step-by-step examples, solutions, and guided practice exercises.

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Rationalize Denominators & Solve Quadratic Equations: Examples & Practice

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  1. 3 7 + 5 2 2 2 5 5 2 2 5 (a) = 2 2 = 10 2 = EXAMPLE 2 Rationalize denominators of fractions. Simplify and (a) (b) SOLUTION

  2. 3 3 7 + 7 + 7 – (b) = 7 – 2 2 2 2 2 2 2 2 21 – 3 = 49 – 7 + 7 – 2 21 – 3 = 47 EXAMPLE 2 Rationalize denominators of fractions. SOLUTION

  3. 3 3 4 + x = 12 + x = + 2 x = EXAMPLE 3 Solve a quadratic equation Solve 3x2 + 5 = 41. 3x2 + 5 = 41 Write original equation. 3x2 = 36 Subtract 5 from each side. x2 = 12 Divide each side by 3. Take square roots of each side. Product property Simplify.

  4. – 2 2 3 3 ? ? 3( )2 + 5 = 41 3( )2 + 5 = 41 2 – 2 3 3 ? ? 3(12) + 5 = 41 3(12) + 5 = 41 41 = 41 41 = 41   EXAMPLE 3 Solve a quadratic equation ANSWER The solutions are and Check the solutions by substituting them into the original equation. 3x2 + 5 = 41 3x2 + 5 = 41

  5. 35 35 35 35 (z + 3)2 = 7 15 z + 3 = + z = –3 + The solutions are –3 + and –3 – EXAMPLE 4 Standardized Test Practice SOLUTION Write original equation. (z + 3)2 = 35 Multiply each side by 5. Take square roots of each side. Subtract 3 from each side.

  6. EXAMPLE 4 Standardized Test Practice ANSWER The correct answer is C.

  7. 17 5 5 19 11 6 11 9 – 6 2 12 21 5 8 4 + 7 – 30 5 3 – 21 – 3 8 – 2 399 2 51 4 21 22 5 6 for Examples 2, 3, and 4 GUIDED PRACTICE GUIDED PRACTICE Simplify the expression. ANSWER ANSWER ANSWER ANSWER ANSWER ANSWER

  8. 7 7 3 3 – 1 4 9 + 8 – – 9 + 32 + 4 74 61 for Examples 2, 3, and 4 GUIDED PRACTICE ANSWER ANSWER

  9. 120 + 2 3 + 4 + 6 for Examples 2, 3, and 4 GUIDED PRACTICE Solve the equation. 5x2 = 80 ANSWER z2 – 7 = 29 ANSWER 3(x – 2)2 = 40 ANSWER

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