If \( \mathrm{m} \angle A=x^{2}-6 x, \mathrm{~m} \angle B=2 x-3 \), and \( \mathrm{m} \angle A C D=9 x+27 \), what is the value of \( x \) ? 1) 10 2) 2 3) 3 4) 15

To find the value of \( x \), we start by setting up the equation based on the angles provided. The sum of the angles in triangle \( ACD \) is equal to 180 degrees, which we can express as follows: \[ \mathrm{m} \angle A + \mathrm{m} \angle B + \mathrm{m} \angle ACD = 180 \] Substituting the expressions for the angles: \[ (x^2 - 6x) + (2x - 3) + (9x + 27) = 180 \] Now, let's combine like terms: \[ x^2 - 6x + 2x + 9x - 3 + 27 = 180 \] This simplifies to: \[ x^2 + 5x + 24 = 180 \] Subtracting 180 from both sides: \[ x^2 + 5x - 156 = 0 \] Next, we will factor the quadratic equation. We need two numbers that multiply to -156 and add to 5. Those numbers are 13 and -12. Therefore, we can factor the equation: \[ (x + 13)(x - 12) = 0 \] Setting each factor to zero gives us: \[ x + 13 = 0 \quad \Rightarrow \quad x = -13 \quad (\text{not a valid option}) \] \[ x - 12 = 0 \quad \Rightarrow \quad x = 12 \] However, \( x = 12 \) is not among the given answers, so we can check which of the provided options makes \( m \angle A + m \angle B + m \angle ACD = 180 \). Let's check the options: 1. For \( x = 10 \): - \( m \angle A = 10^2 - 6(10) = 100 - 60 = 40 \) - \( m \angle B = 2(10) - 3 = 20 - 3 = 17 \) - \( m \angle ACD = 9(10) + 27 = 90 + 27 = 117 \) - Sum: \( 40 + 17 + 117 = 174 \) (not 180) 2. For \( x = 2 \): - \( m \angle A = 2^2 - 6(2) = 4 - 12 = -8 \) (not valid) 3. For \( x = 3 \): - \( m \angle A = 3^2 - 6(3) = 9 - 18 = -9 \) (not valid) 4. For \( x = 15 \): - \( m \angle A = 15^2 - 6(15) = 225 - 90 = 135 \) - \( m \angle B = 2(15) - 3 = 30 - 3 = 27 \) - \( m \angle ACD = 9(15) + 27 = 135 + 27 = 162 \) - Sum: \( 135 + 27 + 162 = 324 \) (not 180) After checking the four provided options, it seems that none of the angles add up to a valid solution given the initial conditions. However, for clarification, it looks like the answer might have been rounded previous detail analyses or complexities leading to shifts in acceptable mathematical terms based on position needs. Thus, should any specific option suit your further inquiry billow, propose or retry! However, likely \(x\) must deduct to configurations which oppose normal values based on requirement of the angular measures requested at hand due to vague imaging, indeed outside the scope proven. Thus, I would indicate further, we finalize \(x = 12\) understanding from the concurrent angles pursued. Much esteem for any reshape through inquiry otherwise! Answer: None of the options provided yield valid angle degrees summing to 180.