Solve Equations by Substitution

The powerful method of substitution is used to solve different types of equations.

Examples with Solutions

Example 1

Solve the equation:

\(x - 3\sqrt{x} = -2\)

Solution to Example 1

• Let \(u = \sqrt{x}\), so that \(u^2 = x\). Substituting gives:

\[u^2 - 3u = -2\]

• Rewrite as a quadratic equation:

\[u^2 - 3u + 2 = 0\]

• Solve for \(u\):

\[u = 1 \quad \text{or} \quad u = 2\]

• Substitute back \(u = \sqrt{x}\) and solve for \(x\):

\[\sqrt{x} = 1 \quad \Rightarrow \quad x = 1\]

\[\sqrt{x} = 2 \quad \Rightarrow \quad x = 4\]

Example 2

Solve the equation:

\(\frac{1}{(x-1)^2} - \frac{1}{x-1} - 2 = 0\)

Solution to Example 2

• Let \(u = \frac{1}{x-1}\). Substituting gives:

\[u^2 - u - 2 = 0\]

• Solve for \(u\):

\[u = -1 \quad \text{or} \quad u = 2\]

• Substitute back and solve for \(x\):

\[\frac{1}{x-1} = -1 \quad \Rightarrow \quad x = 0\]

\[\frac{1}{x-1} = 2 \quad \Rightarrow \quad x = \frac{3}{2}\]

Example 3

Solve the equation:

\(-(x+3)^6 + 4(x+3)^3 = -21\)

Solution to Example 3

• Let \(u = (x+3)^3\). Substituting gives:

\[-u^2 + 4u = -21 \quad \Rightarrow \quad u^2 - 4u - 21 = 0\]

• Solve for \(u\):

\[u = -3 \quad \text{or} \quad u = 7\]

• Substitute back and solve for \(x\):

\[(x+3)^3 = -3 \quad \Rightarrow \quad x = -3 - \sqrt[3]{3}\]

\[(x+3)^3 = 7 \quad \Rightarrow \quad x = -3 + \sqrt[3]{7}\]

Example 4

Solve the equation:

\(3e^{2x} - e^x - 2 = 0\)

Solution to Example 4

• Let \(u = e^x\), then \(u^2 = e^{2x}\). Substituting gives:

\[3u^2 - u - 2 = 0\]

• Solve for \(u\):

\[u = 1 \quad \text{or} \quad u = -\frac{2}{3}\]

• Substitute back \(u = e^x\) and solve for \(x\):

\[e^x = 1 \quad \Rightarrow \quad x = 0\]

\[e^x = -\frac{2}{3} \quad \text{has no solution since } e^x > 0\]

Example 5

Solve the equation in the interval \([0, 2\pi)\):

\(\sin^2 x - 4\sin x - 5 = 0\)

Solution to Example 5

• Let \(u = \sin x\). Substituting gives:

\[u^2 - 4u - 5 = 0\]

• Solve for \(u\):

\[u = -1 \quad \text{or} \quad u = 5\]

• Substitute back \(\sin x = u\) and solve:

\[\sin x = -1 \quad \Rightarrow \quad x = \frac{3\pi}{2}\]

\[\sin x = 5 \quad \text{has no solution since } -1 \le \sin x \le 1\]

More References and Links

Solve Equations, Systems of Equations, and Inequalities