Math Snap

STEP 1

1. The equation $x^2 - 14x + 49 = (x-a)^2$ is a quadratic equation.
2. The left side is already in standard quadratic form.
3. The right side is a perfect square trinomial.
4. Solving the equation involves expanding the right side and equating coefficients.

STEP 2

1. Expand the right side of the equation.
2. Equate the coefficients of the quadratic equation.
3. Solve for the value of $a$.

STEP 3

Expand the right side of the equation. The expression $(x-a)^2$ can be expanded using the formula $(x-a)^2 = x^2 - 2ax + a^2$.
$$(x-a)^2 = x^2 - 2ax + a^2$$

STEP 4

Equate the expanded form of the right side to the left side of the equation. This gives us:
$$x^2 - 14x + 49 = x^2 - 2ax + a^2$$

STEP 5

Since the quadratic terms $x^2$ are the same on both sides, we focus on the linear and constant terms. Equate the coefficients of the linear terms and the constant terms:
1. Linear terms: $-14 = -2a$
2. Constant terms: $49 = a^2$

STEP 6

Solve the equation for the linear terms to find $a$:
$$-14 = -2a$$ $$2a = 14$$ $$a = 7$$

Verify the constant term equation with the found value of $a$:
$$49 = a^2$$ $$49 = 7^2$$ The value of $a$ satisfies both conditions.
The value of $a$ is:
$$\boxed{7}$$