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Solve the equation: sqrt(3)x2 - sqrt(12)x = sqrt(27)

Mathematics

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Answer

x=3 or x=−1x = 3 \text{ or } x = -1

Step-by-step solution

6 steps

  1. Simplify the square roots in the equation.

    3x2−12x=27\sqrt{3}x^2 - \sqrt{12}x = \sqrt{27}

    We can simplify 12\sqrt{12} and 27\sqrt{27}:

    12=4×3=4×3=23\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} 27=9×3=9×3=33\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3}

  2. Substitute the simplified square roots back into the equation.

    3x2−23x=33\sqrt{3}x^2 - 2\sqrt{3}x = 3\sqrt{3}

  3. Rearrange the equation into the standard quadratic form ax2+bx+c=0ax^2 + bx + c = 0.

    3x2−23x−33=0\sqrt{3}x^2 - 2\sqrt{3}x - 3\sqrt{3} = 0

  4. Divide the entire equation by 3\sqrt{3} to simplify.

    3x23−23x3−333=03\frac{\sqrt{3}x^2}{\sqrt{3}} - \frac{2\sqrt{3}x}{\sqrt{3}} - \frac{3\sqrt{3}}{\sqrt{3}} = \frac{0}{\sqrt{3}} x2−2x−3=0x^2 - 2x - 3 = 0

  5. Factor the quadratic equation.

    We need two numbers that multiply to -3 and add to -2. These numbers are -3 and 1.

    (x−3)(x+1)=0(x - 3)(x + 1) = 0

  6. Set each factor equal to zero and solve for xx.

    x−3=0  ⟹  x=3x - 3 = 0 \implies x = 3 x+1=0  ⟹  x=−1x + 1 = 0 \implies x = -1

    The solutions for xx are:

    x=3 or x=−1x = 3 \text{ or } x = -1
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