(х-7)(28-х)=20

Let's solve the equation \((x - 7)(28 - x) = 20\). **Step 1:** Expand the left side: \[ (x - 7)(28 - x) = x \times 28 - x \times x - 7 \times 28 + 7 \times x \] \[ = 28x - x^2 - 196 + 7x \] **Step 2:** Combine like terms: \[ - x^2 + (28x + 7x) - 196 = 20 \] \[ - x^2 + 35x - 196 = 20 \] **Step 3:** Move all terms to one side to set equal to zero: \[ - x^2 + 35x - 196 - 20 = 0 \] \[ - x^2 + 35x - 216 = 0 \] **Step 4:** Multiply through by -1 to simplify: \[ x^2 - 35x + 216 = 0 \] **Step 5:** Solve the quadratic equation \(x^2 - 35x + 216 = 0\). Calculate the discriminant (\(\Delta\)): \[ \Delta = (-35)^2 - 4 \times 1 \times 216 = 1225 - 864 = 361 \] Since \(\sqrt{361} = 19\), the roots are: \[ x = \frac{35 \pm 19}{2} \] Compute each: - \(x = \frac{35 + 19}{2} = \frac{54}{2} = 27\) - \(x = \frac{35 - 19}{2} = \frac{16}{2} = 8\) **Final answer:** \[ \boxed{ x = 8 \quad \text{or} \quad x = 27 } \]