Question

solve each equation. one of each method must be used (factoring, square roots, completing the square and quadratic formula). plan accordingly.

1. $x^2 - 4x = 21$ method: _______________
2. $x^2 - 12x + 12 = 0$ method: _______________
3. $-2x^2 + 12 = 66$ method: ________________
4. $x^2 - 10x = -1$ method: ________________

Explanation:

Problem 10: \( x^2 - 4x = 21 \) (Method: Factoring)

Step 1: Rewrite in standard form

Subtract 21 from both sides: \( x^2 - 4x - 21 = 0 \)

Step 2: Factor the quadratic

Find two numbers that multiply to -21 and add to -4. The numbers are -7 and 3.
So, \( (x - 7)(x + 3) = 0 \)

Step 3: Solve for \( x \)

Set each factor equal to zero:
\( x - 7 = 0 \) or \( x + 3 = 0 \)
\( x = 7 \) or \( x = -3 \)

Step 1: Identify \( a, b, c \)

For \( x^2 - 12x + 12 = 0 \), \( a = 1 \), \( b = -12 \), \( c = 12 \)

Step 2: Apply quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Calculate discriminant: \( b^2 - 4ac = (-12)^2 - 4(1)(12) = 144 - 48 = 96 \)
Then, \( x = \frac{12 \pm \sqrt{96}}{2} = \frac{12 \pm 4\sqrt{6}}{2} = 6 \pm 2\sqrt{6} \)

Step 1: Isolate \( x^2 \)

Subtract 12 from both sides: \( -2x^2 = 66 - 12 = 54 \)
Divide by -2: \( x^2 = \frac{54}{-2} = -27 \)
Wait, this gives a negative number under the square root (no real solutions) or complex solutions. If considering real numbers, no solution. If complex:

Step 2: Take square roots

\( x = \pm \sqrt{-27} = \pm 3i\sqrt{3} \) (complex solutions)

Answer:

\( x = 7 \) or \( x = -3 \)

Problem 11: \( x^2 - 12x + 12 = 0 \) (Method: Quadratic Formula)