Q. \(x^2-49\)

Q: How do I factor x^2-49?

Use difference of squares: x^2-49=(x-7)(x+7).

Q: What are the solutions to x^2-49=0?

Set \,x^2=49, so x=\pm 7. Solutions: x=7 and x=-7.

Q: Can I use the quadratic formula on x^2-49=0?

Here a=1,b=0,c=-49. Then x=\frac{-0\pm\sqrt{0-4(1)(-49)}}{2(1)}=\pm 7.

Q: What is the domain and range of y=x^2-49?

Domain: all real x. Range: y\ge -49 since x^2\ge 0. So minimum value is -49.

Q: Where does the graph of y=x^2-49 cross the x-axis?

Set y=0: x^2-49=0. Thus x=\pm 7, so intercepts are (-7,0) and (7,0).

Q: What is the vertex and axis of symmetry for y=x^2-49?

Vertex at (0,-49). Axis of symmetry is x=0 because the quadratic has no x-term.

Answer

We factor the difference of squares:

\[
x^2 – 49 = x^2 – 7^2 = (x-7)(x+7).
\]

Set it equal to zero:

\[
(x-7)(x+7)=0 \Rightarrow x-7=0 \text{ or } x+7=0.
\]

So the solutions are:

\[
x=7 \text{ or } x=-7.
\]

Detailed Explanation

We are asked to solve the equation that is represented by

\[
x^2 – 49
\]

If the problem is intended to “solve” for \(x\), the usual interpretation is that we set the expression equal to zero:

\[
x^2 – 49 = 0
\]

Step 1: Move the constant term

Add \(49\) to both sides so that the \(x^2\) term is by itself.

\[
x^2 – 49 + 49 = 0 + 49
\]
\[
x^2 = 49
\]

Step 2: Take the square root of both sides

We now solve \(x^2 = 49\). The square root gives two solutions because \(49\) is nonnegative.

\[
x = \pm \sqrt{49}
\]
\[
x = \pm 7
\]

Step 3: Write the final solutions

Therefore, the values of \(x\) are:

\[
x = 7 \quad \text{or} \quad x = -7
\]

See full solution

Graph

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Algebra FAQ

How do I factor \(x^2-49\)?

Use difference of squares: \(x^2-49=(x-7)(x+7)\).

What are the solutions to \(x^2-49=0\)?

Set \(\,x^2=49\), so \(x=\pm 7\). Solutions: \(x=7\) and \(x=-7\).

Can I use the quadratic formula on \(x^2-49=0\)?

Here \(a=1,b=0,c=-49\). Then \(x=\frac{-0\pm\sqrt{0-4(1)(-49)}}{2(1)}=\pm 7\).

What is the domain and range of \(y=x^2-49\)?

Domain: all real \(x\). Range: \(y\ge -49\) since \(x^2\ge 0\). So minimum value is \(-49\).

Where does the graph of \(y=x^2-49\) cross the \(x\)-axis?

Set \(y=0\): \(x^2-49=0\). Thus \(x=\pm 7\), so intercepts are \((-7,0)\) and \((7,0)\).

What is the vertex and axis of symmetry for \(y=x^2-49\)?

Vertex at \((0,-49)\). Axis of symmetry is \(x=0\) because the quadratic has no \(x\)-term.

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