Question

3. Solve for the zeros. Use the Completing the Square method. a) f(x) = x2 – 18x + 53

Answer 1

Answer:

$x=9+ 2\sqrt{7}\\x=9- 2\sqrt{7}$

Step-by-step explanation:

To solve a quadratic equation by completing squares we need to recall these basic algebra relations called the square of a binomial:

$(a+b)^2=a^2+2ab+b^2$

$(a-b)^2=a^2-2ab+b^2$

We can use them from right to left to factor trinomials:

$a^2-2ab+b^2=(a-b)^2$

We need to solve the equation:

$x^2-18x+53=0$

Subtracting 53:

$x^2-18x=-53$

The first two terms of the left side of the equation are the first two terms of the trinomial given in the formula, having:

$a^2=x^2$

Or, equivalently:

$a=x$

Also:

$2ab=-18x$

Since a=x

$2xb=-18x$

Simplifying:

b=-9

We need to complete the square by adding $b^2=81$

Adding 81:

$x^2-18x+81=-53+81$

Factoring and operating:

$(x-9)^2=28$

Since 28=4*7

$(x-9)^2=4*7$

Taking square roots:

$x-9=\pm \sqrt{4*7}$

Separating the square root of 4:

$x-9=\pm 2\sqrt{7}$

Solving:

$x=9\pm 2\sqrt{7}$

The two solutions are:

$x=9+ 2\sqrt{7}\\\\x=9- 2\sqrt{7}$