To solve the equation [tex]\(2x^2 - 7 = 9\)[/tex], one can use algebraic manipulation to isolate the variable [tex]\(x\)[/tex]. Here's a detailed, step-by-step solution:
1. Isolate the quadratic term:
[tex]\[ 2x^2 - 7 = 9 \][/tex]
First, we need to move the constant term to the other side of the equation. To do this, add 7 to both sides:
[tex]\[ 2x^2 - 7 + 7 = 9 + 7 \][/tex]
Simplifying both sides gives:
[tex]\[ 2x^2 = 16 \][/tex]
2. Solve for [tex]\(x^2\)[/tex]:
To isolate [tex]\(x^2\)[/tex], divide both sides by 2:
[tex]\[ \frac{2x^2}{2} = \frac{16}{2} \][/tex]
This simplifies to:
[tex]\[ x^2 = 8 \][/tex]
3. Solve for [tex]\(x\)[/tex]:
To solve for [tex]\(x\)[/tex], take the square root of both sides of the equation. Remember that the square root of a number can be both positive and negative:
[tex]\[ x = \pm \sqrt{8} \][/tex]
4. Simplify the square root:
The square root of 8 can be simplified. By expressing 8 as the product of its prime factors, [tex]\(8 = 4 \times 2\)[/tex], we get:
[tex]\[ \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2} \][/tex]
Thus, the solutions to the equation [tex]\(2x^2 - 7 = 9\)[/tex] are:
[tex]\[ x = \pm 2\sqrt{2} \][/tex]
So, the method I chose here is solving the quadratic equation by isolating the quadratic term and then taking the square root. This method is straightforward and adequate due to the simplicity of the given quadratic equation. It directly leads to the solutions without the need for more advanced techniques.
