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Factorize the quadratic equation 3x² + 8x + 4 = 0, given that P=12 and S=8

Mathematics

ScanSolve AI Answer

Answer

x=−23,−2x = -\frac{2}{3}, -2

Step-by-step solution

6 steps

To solve the quadratic equation 3x2+8x+4=03x^2 + 8x + 4 = 0 by factorization, we need to find two numbers that multiply to the product of the first and last coefficients (P=3×4=12P = 3 \times 4 = 12) and add up to the middle coefficient (S=8S = 8).

  1. Identify the product (P) and sum (S) values.

    The given equation is 3x2+8x+4=03x^2 + 8x + 4 = 0.
    The product P=(3)(4)=12P = (3)(4) = 12.
    The sum S=8S = 8.

  2. Find two numbers that satisfy the product and sum conditions.

    We need two numbers that multiply to 12 and add up to 8.
    The numbers are 2 and 6, since 2×6=122 \times 6 = 12 and 2+6=82 + 6 = 8.

  3. Rewrite the middle term of the equation using these two numbers.

    3x2+2x+6x+4=03x^2 + 2x + 6x + 4 = 0

  4. Factor by grouping.

    Group the terms and factor out the common factor from each pair.

    (3x2+2x)+(6x+4)=0(3x^2 + 2x) + (6x + 4) = 0 x(3x+2)+2(3x+2)=0x(3x + 2) + 2(3x + 2) = 0

  5. Factor out the common binomial.

    (3x+2)(x+2)=0(3x + 2)(x + 2) = 0

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

    3x+2=0orx+2=03x + 2 = 0 \quad \text{or} \quad x + 2 = 0 3x=−2orx=−23x = -2 \quad \text{or} \quad x = -2 x=−23orx=−2x = -\frac{2}{3} \quad \text{or} \quad x = -2

    The solutions for xx are:

    x=−23,−2x = -\frac{2}{3}, -2
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