Solve the following equations: 1. x - 8 = 0 2. x + 4 = 0 3. x + 2 = 0 4. x + z = 0 5. 6x^2 - 9 = 0 6. 2x(x + 3) = 0 7. 2 - 3x + 1 = 0
Understand the Problem
The image shows some handwritten mathematical equations and a textbook cover. We need to understand and solve the handwritten equations.
Answer
$7x - 8 = 0 \implies x = \frac{8}{7}$ $6x^2 - 9 = 3(2x^2 - 3)$ $2x(x+3) = 2x^2 + 6x$ $2 - 3x + 1 = 3 - 3x$ $x + z = 0, z = -7 \implies x = 7$ $x = \frac{3}{2}$
Answer for screen readers
$7x - 8 = 0 \implies x = \frac{8}{7}$ $6x^2 - 9 = 3(2x^2 - 3)$ $2x(x+3) = 2x^2 + 6x$ $2 - 3x + 1 = 3 - 3x$ $x + z = 0, z = -7 \implies x = 7$ $x = \frac{3}{2}$
Steps to Solve
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Solving $7x - 8 = 0$ Add 8 to both sides of the equation: $7x - 8 + 8 = 0 + 8$ $7x = 8$ Divide both sides by 7: $\frac{7x}{7} = \frac{8}{7}$ $x = \frac{8}{7}$
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Simplifying $6x^2 - 9$ We can factor out a 3 from both terms: $6x^2 - 9 = 3(2x^2 - 3)$
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Expanding $2x(x+3)$ Distribute $2x$ to both terms inside the parentheses: $2x(x+3) = 2x \cdot x + 2x \cdot 3 = 2x^2 + 6x$
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Simplifying $2 - 3x + 1$ Combine the constant terms 2 and 1: $2 - 3x + 1 = 3 - 3x$
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Solving $x + z = 0$ and $z = -7$ Substitute $z = -7$ into the equation $x + z = 0$: $x + (-7) = 0$ $x - 7 = 0$ Add 7 to both sides: $x = 7$
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The equation $x = \frac{3}{2}$ is already solved.
$7x - 8 = 0 \implies x = \frac{8}{7}$ $6x^2 - 9 = 3(2x^2 - 3)$ $2x(x+3) = 2x^2 + 6x$ $2 - 3x + 1 = 3 - 3x$ $x + z = 0, z = -7 \implies x = 7$ $x = \frac{3}{2}$
More Information
The problem involves solving and simplifying basic algebraic expressions.
Tips
- Forgetting to distribute when expanding expressions like $2x(x+3)$.
- Incorrectly combining like terms, for example, in $2 - 3x + 1$.
- Making errors when solving for x in simple equations like $7x - 8 = 0$ or $x + z = 0$.
- Not factoring out the greatest common factor when simplifying expressions.
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