Solve the Equation: -2x(3-x)+(x-3)² = 9 for Parameter x

Video Solution

Solution Steps

00:09 Let's solve this problem step by step.

00:16 First, open the parentheses and multiply each factor inside.

00:26 Use the multiplication formulas to expand the parentheses.

00:39 Now, calculate the products from these multiplications.

00:47 Reduce by simplifying where possible.

00:52 Next, group similar terms together.

01:03 Break down each multiplication by using the factor three.

01:12 Take out the common factor from the entire expression.

01:25 Determine when each factor equals zero.

01:29 Then, isolate the unknown variable to find a solution.

01:34 That's one solution. Now, let's use the same method to find another one.

01:39 And there you have it. This is how we solve the problem!

Step-by-Step Solution

To solve the equation $-2x(3-x) + (x-3)^2 = 9$ , follow these steps:

Step 1: Expand each term:
$-2x(3-x) = -6x + 2x^2$ and $(x-3)^2 = x^2 - 6x + 9$ .
Step 2: Substitute the expanded terms into the equation:
$-6x + 2x^2 + x^2 - 6x + 9 = 9$ .
Step 3: Combine like terms:
$(2x^2 + x^2) + (-6x - 6x) + 9 = 9$ .
Step 4: Simplify further:
$3x^2 - 12x + 9 = 9$ .
Step 5: Move all terms to one side to form a quadratic equation:
$3x^2 - 12x + 9 - 9 = 0$ .
Step 6: Simplify the expression:
$3x^2 - 12x = 0$ .
Step 7: Factor out the common term:
$3x(x - 4) = 0$ .
Step 8: Solve for $x$ :
Since $3x = 0$ or $x - 4 = 0$ , we find $x = 0$ or $x = 4$ .

Therefore, the values of $x$ that satisfy the equation are $\mathbf{x = 0}$ and $\mathbf{x = 4}$ .