(
x
−
6
)
(
x
+
4
)
=
(
x
+
12
)
(
−
x
+
3
) . Solve the x.

We need to solve the equation:

\[
(x - 6)(x + 4) = (x + 12)(-x + 3)
\]

### Step 1: Expand both sides

#### Left-hand side:
\[
(x - 6)(x + 4) = x(x + 4) - 6(x + 4) = x^2 + 4x - 6x - 24 = x^2 - 2x - 24
\]

#### Right-hand side:
\[
(x + 12)(-x + 3) = x(-x + 3) + 12(-x + 3) = -x^2 + 3x - 12x + 36 = -x^2 - 9x + 36
\]

### Step 2: Set up the equation

Now we have:

\[
x^2 - 2x - 24 = -x^2 - 9x + 36
\]

### Step 3: Move all terms to one side

Add \(x^2\) and \(9x\) to both sides:

\[
x^2 + x^2 - 2x + 9x - 24 - 36 = 0
\]

Simplifying this:

\[
2x^2 + 7x - 60 = 0
\]

### Step 4: Solve the quadratic equation

We will solve the quadratic equation \(2x^2 + 7x - 60 = 0\) using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \(2x^2 + 7x - 60 = 0\), the coefficients are:
- \(a = 2\)
- \(b = 7\)
- \(c = -60\)

Substituting into the quadratic formula:

\[
x = \frac{-7 \pm \sqrt{7^2 - 4(2)(-60)}}{2(2)}
\]
\[
x = \frac{-7 \pm \sqrt{49 + 480}}{4}
\]
\[
x = \frac{-7 \pm \sqrt{529}}{4}
\]
\[
x = \frac{-7 \pm 23}{4}
\]

### Step 5: Find the solutions

Now we compute the two possible values for \(x\):

1. \(x = \frac{-7 + 23}{4} = \frac{16}{4} = 4\)
2. \(x = \frac{-7 - 23}{4} = \frac{-30}{4} = -7.5\)

### Final Solutions:
The solutions are:

\[
x = 4 \quad \text{or} \quad x = -7.5
\]

Would you like more details or have any questions?

Here are five related problems:
1. Solve \( (x + 5)(x - 7) = (x - 2)(x + 3) \).
2. Solve \( (x + 2)(x - 4) = (x + 1)(x - 5) \).
3. Solve \( x^2 - 4x = 5x - 3 \).
4. Solve \( (2x - 3)(x + 1) = (x + 4)(x - 2) \).
5. Solve \( (x + 6)(x - 3) = (x + 2)(x + 5) \).

**Tip**: When solving quadratic equations, check if factoring is possible before applying the quadratic formula.

(x - 6)(x + 4) = (x + 12)(-x + 3). Solve for x.

Learn how to solve the quadratic equation (x - 6)(x + 4) = (x + 12)(-x + 3) with a comprehensive step-by-step breakdown. This solution covers algebraic concepts, including polynomial expansion and applying the quadratic formula. Suitable for students in Grades 9-10.

Math Problem Statement

( x − 6 ) ( x + 4 )

Solution

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