UpStudy Free Solution:

To factor the quadratic expression \(x^ 2 + 8x = 12\), we first need to rewrite it in standard form:

\[x^ 2 + 8x - 12 = 0\]

Next, we look for two numbers that multiply to \(- 12\) (the constant term) and add up to \(8\) (the coefficient of the \(x\) term).

These two numbers are \(10\) and \(- 2\), since:

\[10 \times ( - 2) = - 12\]

\[10 + ( - 2) = 8\]

Now, we can rewrite the middle term (\(8x\)) using these two numbers:

\[x^ 2 + 10x - 2x - 12 = 0\]

Next, we factor by grouping:

\[( x^ 2 + 10x) + ( - 2x - 12) = 0\]

Factor out the common factors in each group:

\[x( x + 10) - 2( x + 10) = 0\]

Notice that \(( x + 10) \) is a common factor:

\[( x - 2) ( x + 10) = 0\]

So, the factors of \(x^ 2 + 8x - 12\) are \(( x - 2) \) and \(( x + 10) \).

Therefore, the factored form of the original equation \(x^ 2 + 8x = 12\) is:

\[( x - 2) ( x + 10) = 0\]

Supplemental Knowledge

Factoring quadratic equations is a fundamental skill in algebra that involves rewriting a quadratic expression as a product of two binomials. The general form of a quadratic equation is:

\[ax^ 2 + bx + c = 0\]

To factor this, we look for two numbers that multiply to \(ac\) (the product of the coefficient of \(x^ 2\) and the constant term) and add up to \(b\) (the coefficient of \(x\)). Here are the steps involved:

1. Rewrite in Standard Form:

Ensure the quadratic equation is in the form \(ax^ 2 + bx + c = 0\).

2. Identify \(a\), \(b\), and \(c\):

Determine the coefficients from the standard form.

3. Find Two Numbers:

Find two numbers that multiply to \(ac\) and add up to \(b\).

4. Rewrite Middle Term:

Use these two numbers to split the middle term (\(bx\)) into two terms.

5. Factor by Grouping:

Group terms in pairs and factor out common factors from each group.

6. Factor Out Common Binomial:

Factor out the common binomial factor from each group.

For example, let's consider the quadratic equation:

\[x^ 2 + 8x - 12 = 0\]

- Here, \(a = 1\), \(b = 8\), and \(c = - 12\).

- We need two numbers that multiply to \(- 12\) (i.e., \(1 \times - 12\)) and add up to \(8\). These numbers are \(10\) and \(- 2\).

- Rewrite the middle term using these numbers:

\[x^ 2 + 10x - 2x - 12 = 0\]

- Factor by grouping:

\[( x^ 2 + 10x) + ( - 2x - 12) = x( x + 10) - 2( x + 10) = ( x - 2) ( x + 10) \]

Thus, factoring quadratics helps solve equations by finding their roots or solutions.

Understanding quadratic equations will increase both your algebra skills and confidence when solving mathematical issues, whether for homework assignments or exams. Access to reliable resources could make an immense difference to success!