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Factoring is a method used to solve quadratic equations by expressing them as a product of simpler expressions. For the quadratic equation \( x^2 - x - 12 = 0 \), we need to factor the expression \( x^2 - x - 12 \). Our goal is to identify two numbers that multiply to \(-12\) (the constant term) and add up to \(-1\) (the coefficient of the \(x\) term). These magic numbers are \(-4\) and \(+3\). When we use these numbers, the quadratic can be rewritten in a factored form as:

• \((x - 4)(x + 3) = 0\)

This new expression means the equation is now the product of two binomials. Factoring helps simplify the equation, making it easier to solve for the variable \(x\). Always check your factors to ensure they indeed satisfy the original terms of multiplication and addition.

Once the quadratic equation is factored, the next step is finding the roots. Roots are the values of \(x\) that satisfy the quadratic equation, meaning they make the equation true (equal to zero). For the factored equation \((x - 4)(x + 3) = 0\), each binomial expression must be set to zero:

• First, solve \(x - 4 = 0\). This gives the root \(x = 4\).
• Next, solve \(x + 3 = 0\). This gives the root \(x = -3\).

By solving these simple equations, we identify the roots of the quadratic equation as \(x = 4\) and \(x = -3\). These roots are crucial because they represent points where the parabola described by the quadratic equation cuts the x-axis.

Verification is a key step in problem-solving, ensuring the solution is correct. After finding the roots of the quadratic equation \((x = 4\) and \(x = -3\)), we verify them by substituting back into the original equation \(x^2 - x - 12 = 0\).

• For \(x = 4\): Substituting gives \(4^2 - 4 - 12 = 16 - 4 - 12 = 0\), satisfying the equation.
• For \(x = -3\): Substituting gives \((-3)^2 - (-3) - 12 = 9 + 3 - 12 = 0\), also satisfying the equation.

These calculations confirm that both proposed solutions \(x = 4\) and \(x = -3\) are indeed correct. This step is crucial in mathematics to ensure no errors have been made during the factoring and root-finding processes.