1) Verify the equality: 2(x - 3)(x + 7) = 2x^2 + 8x - 42. 2) Solve the equation: 2x^2 + 8x - 42 = 0.
Understand the Problem
The question is asking to verify an algebraic equality and then solve a quadratic equation. In part A, the user needs to show that the two expressions are equal and also find the value of x that satisfies the equation.
Answer
The solutions are $x = -7$ and $x = 3$.
Answer for screen readers
The solutions to the quadratic equation are $x = -7$ and $x = 3$.
Steps to Solve
- Distribute on the left side
Start with the left side of the equation: $$ 2(x - 3)(x + 7) $$ Use the distributive property (FOIL method) on the expression $(x - 3)(x + 7)$: $$ (x - 3)(x + 7) = x^2 + 7x - 3x - 21 = x^2 + 4x - 21 $$ Now multiply by 2: $$ 2(x^2 + 4x - 21) = 2x^2 + 8x - 42 $$
- Verify the equality
Now, compare the left-hand side to the right-hand side: $$ 2(x - 3)(x + 7) = 2x^2 + 8x - 42 $$ Since both sides are equal, the equality is verified.
- Solve the quadratic equation
Next, solve the equation: $$ 2x^2 + 8x - 42 = 0 $$ First, simplify the equation by dividing every term by 2: $$ x^2 + 4x - 21 = 0 $$
- Factor the quadratic
Now factor the quadratic: $$ (x + 7)(x - 3) = 0 $$
- Find the solutions
Set each factor equal to zero: $$ x + 7 = 0 \quad \text{or} \quad x - 3 = 0 $$ This gives us two solutions: $$ x = -7 \quad \text{and} \quad x = 3 $$
The solutions to the quadratic equation are $x = -7$ and $x = 3$.
More Information
The verification shows that both sides of the equation $2(x - 3)(x + 7) = 2x^2 + 8x - 42$ are equal. The factored form of the quadratic $x^2 + 4x - 21$ leads us to the solutions $x = -7$ and $x = 3$, which correspond to the points where the parabola intersects the x-axis.
Tips
- Not distributing correctly: Ensure to use the distributive property properly when expanding expressions.
- Forgetting to simplify: Always check if the quadratic can be simplified before factoring or using the quadratic formula.
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