Question2. Solve the equation by completing the square $x^{2}-12 x+2=-9$

Studdy Solution

STEP 1

1. The equation $x^2 - 12x + 2 = -9$ is a quadratic equation.
2. Completing the square is a method that involves creating a perfect square trinomial from the quadratic expression.
3. The equation will be solved for real values of $x$ .

STEP 2

1. Move the constant term to the other side of the equation.
2. Complete the square on the left-hand side.
3. Solve for $x$ by isolating the variable.

STEP 3

Move the constant term on the left-hand side to the right-hand side of the equation:
$x^2 - 12x + 2 = -9$
Subtract 2 from both sides:
$x^2 - 12x = -9 - 2$
Simplify the right-hand side:
$x^2 - 12x = -11$

STEP 4

To complete the square, take half of the coefficient of $x$ , square it, and add it to both sides of the equation. The coefficient of $x$ is $-12$ .
Calculate half of $-12$ :
$\frac{-12}{2} = -6$
Square $-6$ :
$(-6)^2 = 36$
Add 36 to both sides of the equation:
$x^2 - 12x + 36 = -11 + 36$

STEP 5

The left-hand side is now a perfect square trinomial. Rewrite it as a squared binomial:
$(x - 6)^2 = 25$

STEP 6

Solve for $x$ by taking the square root of both sides. Remember to consider both the positive and negative square roots:
$x - 6 = \pm \sqrt{25}$
$x - 6 = \pm 5$

STEP 7

Solve for $x$ by isolating the variable for both cases:
Case 1: $x - 6 = 5$
$x = 5 + 6$
$x = 11$
Case 2: $x - 6 = -5$
$x = -5 + 6$
$x = 1$
The solutions to the equation are:
$x = 11 \quad \text{and} \quad x = 1$

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