Solve the following quadratic equation for all values of x in simplest form: 2(x² - 6) - 1 = -5.
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. We will isolate the variable x by rearranging the equation and applying the quadratic formula if necessary.
Answer
The solutions are $x = 2$ and $x = -2$.
Answer for screen readers
The solutions to the equation are:
$$ x = 2 \quad \text{and} \quad x = -2 $$
Steps to Solve
- Distributing the equation
Distribute the 2 in the equation:
$$ 2(x^2 - 6) - 1 = -5 $$
This simplifies to:
$$ 2x^2 - 12 - 1 = -5 $$
- Combine like terms on the left side
Combine the constants on the left side:
$$ 2x^2 - 13 = -5 $$
- Rearranging the equation
Add 13 to both sides to isolate the quadratic expression:
$$ 2x^2 = 8 $$
- Dividing by the coefficient of x²
Divide both sides by 2:
$$ x^2 = 4 $$
- Taking the square root
Take the square root of both sides, remembering both the positive and negative solutions:
$$ x = \pm 2 $$
The solutions to the equation are:
$$ x = 2 \quad \text{and} \quad x = -2 $$
More Information
This problem involves solving a quadratic equation and recognizing that, when simplified, it can be easily solved using square roots. The solutions correspond to the x-intercepts of the quadratic function represented by the equation.
Tips
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