Question

To solve the given equation, we need to factorize it:

$x^{2}+x-6=0$

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

Setting each factor to zero gives:

$x+3=0$ or $x-2=0$

$x=-3$ or $x=2$

Therefore, the solutions to the equation are $x=-3$ and $x=2$.

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Answer

To solve the quadratic equation $x^2 + x - 6 = 0$, we will use the method of middle term factorization.

Steps to Solve:

  1. Identify the quadratic polynomial: The given equation is: $$ x^2 + x - 6 = 0 $$

  2. Factorize the middle term: We need two numbers whose sum is the coefficient of the middle term ($1$) and whose product is the constant term ($-6$). These numbers are $3$ and $-2$ because: $$ 3 + (-2) = 1 \quad \text{and} \quad 3 \cdot (-2) = -6 $$

  3. Rewrite the quadratic equation: Using these numbers, we split the middle term: $$ x^2 + 3x - 2x - 6 = 0 $$

  4. Group the terms: Grouping the terms to factor by grouping: $$ (x^2 + 3x) + (-2x - 6) = 0 $$

  5. Factor each group: Factoring out the common factors in each group: $$ x(x + 3) - 2(x + 3) = 0 $$

  6. Combine the factors: Now, factor out the common binomial $(x + 3)$: $$ (x + 3)(x - 2) = 0 $$

  7. Solve for $x$: Set each factor equal to $0$ and solve for $x$: $$ x + 3 = 0 \quad \text{or} \quad x - 2 = 0 $$ Hence, $$ x = -3 \quad \text{or} \quad x = 2 $$

The solutions to the equation are $x = -3$ and $x = 2$.


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