Solve for x x+ square root of x=72

Problem

$x+\sqrt{x}=72$

Solution

1. Substitute a new variable to simplify the equation by letting $u=\sqrt{x}$ which implies $u^2=x$

2. Rewrite the original equation in terms of $u$ to form a quadratic equation.

$u^2+u=72$

3. Rearrange the equation into standard quadratic form by subtracting 72 from both sides.

$u^2+u-72=0$

4. Factor the quadratic expression by finding two numbers that multiply to $-72$ and add to $1$

$(u+9)(u-8)=0$

5. Solve for $u$ by setting each factor to zero.

$u=-9$

$u=8$

6. Back-substitute $u=\sqrt{x}$ to find the values of $x$ Since the principal square root $\sqrt{x}$ cannot be negative, we discard $u=-9$

$\sqrt{x}=8$

7. Square both sides of the valid equation to solve for $x$

$x=8^2$

$x=64$

Final Answer

$x=64$

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