Questions: Solve the following equation by making an appropriate substitution. (x+5)^2+17(x+5)+72=0

Solution Steps

To solve the given equation \((x+5)^{2}+17(x+5)+72=0\), we can make a substitution to simplify it. Let \( u = x + 5 \). This transforms the equation into a standard quadratic form: \( u^2 + 17u + 72 = 0 \). We can then solve this quadratic equation for \( u \) using the quadratic formula. Once we find the values of \( u \), we can substitute back to find the corresponding values of \( x \).

To simplify the equation \((x+5)^{2}+17(x+5)+72=0\), we make the substitution \( u = x + 5 \). This transforms the equation into a standard quadratic form:

\[ u^2 + 17u + 72 = 0 \]

We solve the quadratic equation \( u^2 + 17u + 72 = 0 \) using the quadratic formula:

\[ u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

where \( a = 1 \), \( b = 17 \), and \( c = 72 \). Substituting these values, we get:

\[ u = \frac{-17 \pm \sqrt{17^2 - 4 \cdot 1 \cdot 72}}{2 \cdot 1} \]

\[ u = \frac{-17 \pm \sqrt{289 - 288}}{2} \]

\[ u = \frac{-17 \pm \sqrt{1}}{2} \]

\[ u = \frac{-17 \pm 1}{2} \]

This gives us the solutions:

\[ u_1 = \frac{-17 + 1}{2} = -8 \]

\[ u_2 = \frac{-17 - 1}{2} = -9 \]

Substitute back to find \( x \) using \( u = x + 5 \):

For \( u_1 = -8 \):

\[ x + 5 = -8 \implies x = -8 - 5 = -13 \]

For \( u_2 = -9 \):

\[ x + 5 = -9 \implies x = -9 - 5 = -14 \]

Final Answer