Q. Solve the following quadratic equation for all values of \(x\) in simplest form: \(2(x^{2}-6)-1=-5\).

Q: What are the solutions of 2(x^2-6)-1=-5?.

Simplify: 2(x^2-6)-1=-5\Rightarrow 2x^2-13=-5\Rightarrow 2x^2-8=0\Rightarrow x^2-4=0. Factor or take roots: x=\pm 2.

Q: How do you simplify the equation step-by-step?

Expand and combine: 2(x^2-6)-1=2x^2-12-1=2x^2-13. Move -5: 2x^2-13+5=0\Rightarrow 2x^2-8=0. Divide by 2: x^2-4=0.

Q: Can I factor instead of using roots?

Yes. x^2-4=(x-2)(x+2). Set each factor zero: x-2=0 \Rightarrow x=2, x+2=0 \Rightarrow x=-2.

Q: How would the quadratic formula work here?.

For x^2-4=0 with a=1,b=0,c=-4: x=\frac{-0\pm\sqrt{0^2-4(1)(-4)}}{2(1)}=\frac{\pm\sqrt{16}}{2}=\pm 2.

Q: How can I check the solutions quickly?

Substitute: for x=2, 2(4-6)-1=2(-2)-1=-5. For x=-2, same because x^2 same. Both satisfy the original equation.

Q: What is the graph interpretation?

Let f(x)=2(x^2-6)-1=2x^2-13. The horizontal line y=-5 intersects the parabola at x=\pm 2. The parabola vertex is at (0,-13)..

Answer

Start with \(2(x^2-6)-1=-5\). Then \(2x^2-12-1=-5\), so \(2x^2-13=-5\). Add 5: \(2x^2-8=0\). Divide by 2: \(x^2-4=0\). Thus \(x^2=4\) and \(x=\pm 2\).

Final result: \(x=2\) or \(x=-2\).

Detailed Explanation

Step-by-step solution

Write the original equation:\(2(x^{2}-6)-1=-5\)
Apply the distributive property (multiply 2 through the parentheses):\(2\cdot x^{2}-2\cdot 6 – 1 = -5\)
So we get

\(2x^{2}-12-1=-5\)
Combine like terms on the left side (add -12 and -1):\(-12-1=-13\), therefore
\(2x^{2}-13=-5\)
Move all terms to one side to obtain a standard quadratic form:Add 5 to both sides:
\(2x^{2}-13+5=0\)

Combine -13 and +5:

\(2x^{2}-8=0\)
Simplify by dividing both sides by 2 (or factor out 2):Divide by 2:
\(x^{2}-4=0\)
Factor the difference of squares:\(x^{2}-4=(x-2)(x+2)\)
Set each factor equal to zero:

\(x-2=0\) or \(x+2=0\)

So

\(x=2\) or \(x=-2\)
Check the solutions (optional verification):For \(x=2\): \(2(2^{2}-6)-1=2(4-6)-1=2(-2)-1=-4-1=-5\) ✓
For \(x=-2\): \(2((-2)^{2}-6)-1=2(4-6)-1=2(-2)-1=-4-1=-5\) ✓

Final answer: \(x=2\) or \(x=-2\)

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Algebra FAQs

What are the solutions of \(2(x^2-6)-1=-5\)?.

Simplify: \(2(x^2-6)-1=-5\Rightarrow 2x^2-13=-5\Rightarrow 2x^2-8=0\Rightarrow x^2-4=0\). Factor or take roots: \(x=\pm 2\).

How do you simplify the equation step-by-step?

Expand and combine: \(2(x^2-6)-1=2x^2-12-1=2x^2-13\). Move \(-5\): \(2x^2-13+5=0\Rightarrow 2x^2-8=0\). Divide by 2: \(x^2-4=0\).

Can I factor instead of using roots?

Yes. \(x^2-4=(x-2)(x+2)\). Set each factor zero: \(x-2=0 \Rightarrow x=2\), \(x+2=0 \Rightarrow x=-2\).

How would the quadratic formula work here?.

For \(x^2-4=0\) with \(a=1,b=0,c=-4\): \(x=\frac{-0\pm\sqrt{0^2-4(1)(-4)}}{2(1)}=\frac{\pm\sqrt{16}}{2}=\pm 2\).

How can I check the solutions quickly?

Substitute: for \(x=2\), \(2(4-6)-1=2(-2)-1=-5\). For \(x=-2\), same because \(x^2\) same. Both satisfy the original equation.

Are there any extraneous solutions from the steps?.

What is the graph interpretation?

Let \(f(x)=2(x^2-6)-1=2x^2-13\). The horizontal line \(y=-5\) intersects the parabola at \(x=\pm 2\). The parabola vertex is at \((0,-13)\)..

What is the domain of the equation?

The domain is all real numbers, since polynomials are defined for every real \(x\).

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