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Factor completely.

3x^(2)+4x-7
Answer:

Factor completely.\newline3x2+4x7 3 x^{2}+4 x-7 \newlineAnswer:

Full solution

Q. Factor completely.\newline3x2+4x7 3 x^{2}+4 x-7 \newlineAnswer:
  1. Identify Quadratic Expression: Identify the quadratic expression to be factored.\newlineThe given expression is 3x2+4x73x^2 + 4x - 7, which is a quadratic expression in the form ax2+bx+cax^2 + bx + c, where a=3a = 3, b=4b = 4, and c=7c = -7.
  2. Check Factorization Possibility: Check if the quadratic expression can be factored using simple methods.\newlineSince the expression does not factor neatly using methods like factoring by grouping or simple trinomial factoring, we will need to use the quadratic formula to check for possible rational roots or determine that it cannot be factored over the integers.
  3. Apply Quadratic Formula: Apply the quadratic formula to find the roots of the equation 3x2+4x7=03x^2 + 4x - 7 = 0.\newlineThe quadratic formula is x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. For our expression, a=3a = 3, b=4b = 4, and c=7c = -7. Let's calculate the discriminant b24acb^2 - 4ac first.\newlineDiscriminant = b24ac=(4)24(3)(7)=16+84=100b^2 - 4ac = (4)^2 - 4(3)(-7) = 16 + 84 = 100.
  4. Calculate Roots: Since the discriminant is a perfect square, calculate the roots using the quadratic formula.\newlinex=4±10023x = \frac{{-4 \pm \sqrt{100}}}{{2 \cdot 3}}\newlinex=4±106x = \frac{{-4 \pm 10}}{6}\newlineThis gives us two possible roots: x=1046x = \frac{{10 - 4}}{6} and x=1046x = \frac{{-10 - 4}}{6}.\newlinex=66x = \frac{6}{6} or x=146x = \frac{-14}{6}\newlinex=1x = 1 or x=73x = -\frac{7}{3}
  5. Write Factored Form: Write the factored form using the roots found.\newlineThe factored form of the quadratic expression is (x1)(3x+7)(x - 1)(3x + 7).

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