Question Solve for $a$ in the equation $x^{2}-18x+81=(x-a)^{2}$ .

Studdy Solution

STEP 1

Assumptions1. The equation given is $x^{}-18 x+81=(x-a)^{}$ . We are asked to find the value of $a$
3. The equation is quadratic in nature4. The equation is valid for all real numbers

STEP 2

We can start by expanding the right side of the equation $(x-a)^{2}$ .
$(x-a)^{2} = x^{2} -2ax + a^{2}$

STEP 3

Now, we substitute the expanded form of $(x-a)^{2}$ back into the original equation.
$x^{2}-18 x+81=x^{2} -2ax + a^{2}$

STEP 4

Next, we can subtract $x^{2}$ from both sides of the equation to simplify it.
$-18 x+81=-2ax + a^{2}$

STEP 5

We can rearrange the equation to group like terms together.
$2ax -18x + a^{2} -81 =0$

STEP 6

Next, we can factor out an $x$ from the first two terms on the left side of the equation.
$x(2a -18) + a^{2} -81 =0$

STEP 7

Since the equation is quadratic, the coefficients of $x$ should be equal. So, we can set $2a -18$ equal to $-18$ .
$2a -18 = -18$

STEP 8

olving for $a$ , we add $18$ to both sides of the equation.
$2a =0$

STEP 9

Finally, we divide both sides of the equation by $2$ to solve for $a$ .
$a =$ So, the value of $a$ that satisfies the equation is $$.

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