Solve the following quadratic equation for all values of x in simplest form: 2(x + 4)² + 15 = 17.

Question image

Understand the Problem

The question is asking to solve the quadratic equation given in the image for all values of x and to express the solutions in their simplest form.

Answer

$x = -3 \quad \text{and} \quad x = -5$
Answer for screen readers

The solutions to the equation $2(x + 4)^2 + 15 = 17$ are $x = -3$ and $x = -5$.

Steps to Solve

  1. Isolate the quadratic term

Start by subtracting 15 from both sides of the equation to isolate the quadratic term: $$ 2(x + 4)^2 + 15 - 15 = 17 - 15 $$

This simplifies to: $$ 2(x + 4)^2 = 2 $$

  1. Divide by the coefficient of the square

Next, divide both sides by 2: $$ \frac{2(x + 4)^2}{2} = \frac{2}{2} $$

This simplifies to: $$ (x + 4)^2 = 1 $$

  1. Take the square root of both sides

Next, take the square root of both sides. Remember to consider both the positive and negative roots: $$ x + 4 = \pm 1 $$

  1. Solve for x

Now, solve each equation separately.

For the positive root: $$ x + 4 = 1 \ x = 1 - 4 \ x = -3 $$

For the negative root: $$ x + 4 = -1 \ x = -1 - 4 \ x = -5 $$

  1. List the solutions

The solutions to the quadratic equation are: $$ x = -3 \quad \text{and} \quad x = -5 $$

The solutions to the equation $2(x + 4)^2 + 15 = 17$ are $x = -3$ and $x = -5$.

More Information

Quadratic equations often have two solutions, and taking the square root introduces both positive and negative possibilities. Here, we confirmed that both roots are valid solutions.

Tips

  • Forgetting to consider both the positive and negative square roots when taking the square root of both sides.
  • Misapplying the distributive property or simplifying incorrectly when manipulating the equation.

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