Problem 35 Solve each equation. $$ 2 x^... [FREE SOLUTION]

Chapter 5: Problem 35

Solve each equation. $$ 2 x^{2}-12-4 x=x^{2}-3 x $$

Short Answer

Expert verified

x = 4 or x = -3

Step by step solution

- Simplify Both Sides

Combine like terms on both sides of the equation. Start by subtracting the right side from the left side:\[2x^2 - 12 - 4x - (x^2 - 3x) = 0\]This simplifies to:\[2x^2 - x^2 - 4x + 3x - 12 = 0\]

- Combine Like Terms

Combine the terms to get a standard quadratic equation format. This gives:\[x^2 - x - 12 = 0\]

- Factor the Quadratic Equation

Factor the quadratic equation. Look for two numbers that multiply to -12 and add to -1:\[(x - 4)(x + 3) = 0\]

- Solve for x

Set each factor equal to zero and solve for x:\[x - 4 = 0 \implies x = 4\]and\[x + 3 = 0 \implies x = -3\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combining Like Terms

When working with equations, you often need to simplify them by combining like terms. 'Like terms' are terms that have the same variables raised to the same power. In this exercise, we start with the equation: \[2x^2 - 12 - 4x = x^2 - 3x\] To simplify, we subtract the right side from the left side: \[2x^2 - 12 - 4x - (x^2 - 3x) = 0\] This way, we can more easily combine terms. By rearranging and combining, we get: \[2x^2 - x^2 - 4x + 3x - 12 = 0\] Which further simplifies to: \[x^2 - x - 12 = 0\] Always be vigilant with signs (positive or negative) and ensure all like terms are combined correctly.

Factoring Quadratic Equations

Once the quadratic equation is in standard form, like \[x^2 - x - 12 = 0\] we need to factor it. Factoring means finding two numbers that multiply together to give the constant term (-12 in this case) and add up to the coefficient of the middle term (which is -1 here). For \[x^2 - x - 12\] we need two numbers that multiply to -12 and sum up to -1. These numbers are -4 and 3, so we can rewrite the quadratic as: \[(x - 4)(x + 3) = 0\] Factoring can be challenging, but practice and knowing multiplication tables well can make it easier.

Zero Product Property

The Zero Product Property is a principle that states if the product of two factors is zero, then at least one of the factors must be zero. With our factored equation: \[(x - 4)(x + 3) = 0\] we apply the Zero Product Property, setting each factor to zero and solving for x: \[x - 4 = 0 \text{ or } x + 3 = 0\] This gives us: \[x = 4 \text{ or } x = -3\] This means the original quadratic equation \[2x^2 - 12 - 4x = x^2 - 3x\] has two solutions: x = 4 and x = -3. This property is vital for solving polynomial equations of any degree.

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91影视

Solve each equation. $$ 2 x^{2}-12-4 x=x^{2}-3 x $$

Short Answer

Step by step solution

- Simplify Both Sides

- Combine Like Terms

- Factor the Quadratic Equation

- Solve for x

Key Concepts

Combining Like Terms

Factoring Quadratic Equations

Zero Product Property

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