Solve the following quadratic equation for all values of x in simplest form: 2(x^2 - 6) - 1 = -5.
Understand the Problem
The question is asking to solve the given quadratic equation for all values of x and to present the solutions in their simplest form.
Answer
The solutions are \( x = 2 \) and \( x = -2 \).
Answer for screen readers
The solutions for the equation are ( x = 2 ) and ( x = -2 ).
Steps to Solve
- Expand the Equation
Start by distributing the 2 in the equation:
$$ 2(x^2 - 6) - 1 = -5 $$
This simplifies to:
$$ 2x^2 - 12 - 1 = -5 $$
So the equation becomes:
$$ 2x^2 - 13 = -5 $$
- Isolate the Quadratic Term
Next, add 13 to both sides to isolate the quadratic term:
$$ 2x^2 - 13 + 13 = -5 + 13 $$
Which simplifies to:
$$ 2x^2 = 8 $$
- Divide by the Coefficient of (x^2)
Now, divide both sides by 2 to solve for (x^2):
$$ x^2 = \frac{8}{2} $$
This gives:
$$ x^2 = 4 $$
- Take the Square Root
Finally, take the square root of both sides:
$$ x = \pm \sqrt{4} $$
This results in:
$$ x = \pm 2 $$
The solutions for the equation are ( x = 2 ) and ( x = -2 ).
More Information
These solutions represent the points where the quadratic function crosses the x-axis. Quadratic equations can have two real solutions, one real solution, or no real solutions depending on the discriminant.
Tips
- Not following the order of operations: Ensure to properly handle addition and subtraction when rearranging the equation.
- Ignoring negative roots: Remember to consider both the positive and negative square roots when solving for (x).
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