What is the solution set for the equation $\sqrt{5 x + 29} = x + 3$ $sqrt(5x+29)=x+3$ ?

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Answer 1 · 2017-01-02T14:35:22

There is no real solution.

By convention ( definition or tradition or practice ),

$\sqrt{a} \geq 0$ $sqrt(a) >=0$ .

Also, $a \geq 0$ $a >=0$ for the radical to be real.

Here,

$\sqrt{5 x + 3} = (x + 3) \geq 0$ $sqrt(5x+3)=(x+3) >=0$ , giving $x ≻ - 3 .$ $x >--3.$

Also, $a = 5 x + 3 \geq 0$ $a = 5x + 3 >=0$ , giving $x \geq - \frac{3}{5}$ $x >=-3/5$ that satisfies $x ≻ - 3 .$ $x >--3.$

Squaring both sides,

${(x + 3)}^{2} = 5 x + 3$ $(x+3)^2=5x+3$ , giving

$x^{2} + x + 6 = 0$ $x^2+x+6=0$ .

The zeros are complex.

So, there is no real solution.

In the Socratic graph, see that the graph does not cut the x-axis,

Look at the dead end at $x = - \frac{3}{5}$ $x = -3/5$ .

graph{sqrt(5x+3)-x-3 [-15.06, 15.07, -7.53, 7.53]}

What is the solution set for the equation √5x+29=x+3sqrt(5x+29)=x+3?