Answer :
To solve the quadratic expression [tex]\(x^2 + 7x + 12\)[/tex], we start by recognizing that it is a quadratic equation of the form [tex]\(ax^2 + bx + c\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = 7\)[/tex], and [tex]\(c = 12\)[/tex].
Next, we look for two numbers that multiply to [tex]\(ac\)[/tex] (which is [tex]\(1 \times 12 = 12\)[/tex]) and add up to [tex]\(b\)[/tex] (which is [tex]\(7\)[/tex]). These numbers are [tex]\(3\)[/tex] and [tex]\(4\)[/tex], since:
[tex]\[ 3 \cdot 4 = 12 \][/tex]
[tex]\[ 3 + 4 = 7 \][/tex]
We can then factor the quadratic expression as follows:
[tex]\[ x^2 + 7x + 12 = (x + 3)(x + 4) \][/tex]
To verify that this factorization is correct, we can expand [tex]\( (x + 3)(x + 4) \)[/tex] and check if it equals the original quadratic expression:
[tex]\[ (x + 3)(x + 4) = x(x + 4) + 3(x + 4) \][/tex]
[tex]\[ = x^2 + 4x + 3x + 12 \][/tex]
[tex]\[ = x^2 + 7x + 12 \][/tex]
This confirms that the factorization is accurate. Thus, the given quadratic expression [tex]\( x^2 + 7x + 12 \)[/tex] can be expressed as the product of two binomials:
[tex]\[ x^2 + 7x + 12 = (x + 3)(x + 4) \][/tex]
That is the detailed, step-by-step solution for the given quadratic expression.
Next, we look for two numbers that multiply to [tex]\(ac\)[/tex] (which is [tex]\(1 \times 12 = 12\)[/tex]) and add up to [tex]\(b\)[/tex] (which is [tex]\(7\)[/tex]). These numbers are [tex]\(3\)[/tex] and [tex]\(4\)[/tex], since:
[tex]\[ 3 \cdot 4 = 12 \][/tex]
[tex]\[ 3 + 4 = 7 \][/tex]
We can then factor the quadratic expression as follows:
[tex]\[ x^2 + 7x + 12 = (x + 3)(x + 4) \][/tex]
To verify that this factorization is correct, we can expand [tex]\( (x + 3)(x + 4) \)[/tex] and check if it equals the original quadratic expression:
[tex]\[ (x + 3)(x + 4) = x(x + 4) + 3(x + 4) \][/tex]
[tex]\[ = x^2 + 4x + 3x + 12 \][/tex]
[tex]\[ = x^2 + 7x + 12 \][/tex]
This confirms that the factorization is accurate. Thus, the given quadratic expression [tex]\( x^2 + 7x + 12 \)[/tex] can be expressed as the product of two binomials:
[tex]\[ x^2 + 7x + 12 = (x + 3)(x + 4) \][/tex]
That is the detailed, step-by-step solution for the given quadratic expression.
