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3=sqrt(2x^(2)-5x+39)-x What is the product of all solutions to the given equation?

3=2x2−5x+39−x 3=\sqrt{2 x^{2}-5 x+39}-x \newlineWhat is the product of all solutions to the given equation?

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Relationship between squares and square rootsright chevron icon

Full solution

Q. 3=2x2−5x+39−x 3=\sqrt{2 x^{2}-5 x+39}-x \newlineWhat is the product of all solutions to the given equation?
  1. Square Both Sides: Now, square both sides to eliminate the square root. \newline(3+x)2=(2x2−5x+39)2(3 + x)^2 = (\sqrt{2x^{2} - 5x + 39})^2\newline9+6x+x2=2x2−5x+399 + 6x + x^2 = 2x^2 - 5x + 39
  2. Move Terms to One Side: Next, move all terms to one side to set the equation to zero.\newline0=2x2−5x+39−9−6x−x20 = 2x^2 - 5x + 39 - 9 - 6x - x^2\newline0=x2−11x+300 = x^2 - 11x + 30
  3. Factor Quadratic Equation: Now, factor the quadratic equation.\newline0=(x−5)(x−6)0 = (x - 5)(x - 6)
  4. Find Solutions: Find the solutions by setting each factor equal to zero.\newlinex−5=0x - 5 = 0 or x−6=0x - 6 = 0\newlineSo, x=5x = 5 or x=6x = 6
  5. Find Product: To find the product of the solutions, multiply them together.\newlineProduct = 5×65 \times 6\newlineProduct = 3030

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