(x+3)(x-4)+(x-1)(x+1)=10

Jasonbradley

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Jun 16, 2024

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Solving the Equation: (x+3)(x-4) + (x-1)(x+1) = 10

This article will guide you through the steps of solving the equation (x+3)(x-4) + (x-1)(x+1) = 10. We'll start by expanding the equation and then use algebraic manipulation to find the values of x that satisfy the equation.

Expanding the Equation

We'll start by expanding the products on the left-hand side of the equation using the distributive property (or FOIL method):

• (x+3)(x-4) = x(x-4) + 3(x-4) = x² - 4x + 3x - 12 = x² - x - 12
• (x-1)(x+1) = x(x+1) - 1(x+1) = x² + x - x - 1 = x² - 1

Now, we can substitute these expanded expressions back into the original equation:

(x² - x - 12) + (x² - 1) = 10

Simplifying the Equation

Combining like terms on the left-hand side, we get:

2x² - x - 13 = 10

To solve for x, we need to set the equation to zero. Subtracting 10 from both sides gives us:

2x² - x - 23 = 0

Solving the Quadratic Equation

We now have a quadratic equation in the form ax² + bx + c = 0. There are a couple of ways to solve this:

• Factoring: If the quadratic equation can be factored, we can set each factor equal to zero and solve for x. Unfortunately, in this case, the equation does not factor easily.
• Quadratic Formula: The quadratic formula is a more general approach to solving quadratic equations. The formula is:

x = (-b ± √(b² - 4ac)) / 2a

In our equation, a = 2, b = -1, and c = -23. Substituting these values into the quadratic formula gives us:

x = (1 ± √((-1)² - 4 * 2 * -23)) / (2 * 2)

x = (1 ± √185) / 4

Therefore, the solutions to the equation (x+3)(x-4) + (x-1)(x+1) = 10 are:

x = (1 + √185) / 4 and x = (1 - √185) / 4