Questions: Solve the equation. (x-8)^2-3(x-8)-28=0

Solution Steps

To solve the given equation, we can use a substitution method. Let \( y = x - 8 \). This transforms the equation into a quadratic equation in terms of \( y \). We can then solve this quadratic equation using the quadratic formula. Once we find the values of \( y \), we can substitute back to find the corresponding values of \( x \).

We start by substituting \( y = x - 8 \) into the original equation \((x-8)^{2}-3(x-8)-28=0\). This gives us the equation in terms of \( y \): \[ y^2 - 3y - 28 = 0 \]

The equation \( y^2 - 3y - 28 = 0 \) is a quadratic equation. We solve it using the quadratic formula: \[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1 \), \( b = -3 \), and \( c = -28 \).

Calculating the discriminant: \[ b^2 - 4ac = (-3)^2 - 4 \times 1 \times (-28) = 9 + 112 = 121 \]

The solutions for \( y \) are: \[ y = \frac{3 \pm \sqrt{121}}{2} = \frac{3 \pm 11}{2} \]

This gives us two solutions for \( y \): \[ y_1 = \frac{3 + 11}{2} = 7, \quad y_2 = \frac{3 - 11}{2} = -4 \]

Substitute back \( y = x - 8 \) to find \( x \):

• For \( y_1 = 7 \): \[ x - 8 = 7 \implies x = 15 \]
• For \( y_2 = -4 \): \[ x - 8 = -4 \implies x = 4 \]

Final Answer