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Extraneous Solutions Calculator

Get detailed solutions to your math problems with our Extraneous Solutions step-by-step calculator. Practice your math skills and learn step by step with our math solver. Check out all of our online calculators here.

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Example

Solved Problems

Difficult Problems

1

Here, we show you a step-by-step solved example of extraneous solutions. This solution was automatically generated by our smart calculator:

$\sqrt{x}+x=6$
2

Move the term with the square root to the left side of the equation, and all other terms to the right side. Remember to change the signs of each term

$\sqrt{x}=6-x$

Removing the variable's exponent raising both sides of the equation to the power of $2$

$\left(\sqrt{x}\right)^2=\left(6-x\right)^2$

Cancel exponents $\frac{1}{2}$ and $2$

$x=\left(6-x\right)^2$
3

Removing the variable's exponent raising both sides of the equation to the power of $2$

$x=\left(6-x\right)^2$

Square of the first term: $\left(6\right)^2 = .

Double product of the first by the second: $2\left(6\right)\left(-x\right) = .

Square of the second term: $\left(-x\right)^2 =

Expand $\left(6-x\right)^2$

$x=6^2+2\cdot 6\cdot -1x+\left(-x\right)^2$

Multiply $2$ times $6$

$x=6^2+12\cdot -1x+\left(-x\right)^2$

Multiply $12$ times $-1$

$x=6^2-12x+\left(-x\right)^2$

Calculate the power $6^2$

$x=36-12x+\left(-x\right)^2$

Simplify $\left(-x\right)^2$

$x=36-12x+x^2$

Expand $\left(6-x\right)^2$

$6^2+2\cdot 6\cdot -1x+\left(-x\right)^2$

Simplify $\left(-x\right)^2$

$6^2+2\cdot 6\cdot -1x+x^2$
4

Expand $\left(6-x\right)^2$

$x=36-12x+x^2$
5

Group the terms of the equation by moving the terms that have the variable $x$ to the left side, and those that do not have it to the right side

$x+12x-x^2=36$
6

Combining like terms $x$ and $12x$

$13x-x^2=36$
7

Rewrite the equation

$-x^2+13x-36=0$
8

Factor the trinomial by $-1$ for an easier handling

$-\left(x^2-13x+36\right)=0$
9

Factor the trinomial $-\left(x^2-13x+36\right)$ finding two numbers that multiply to form $36$ and added form $-13$

$\begin{matrix}\left(-4\right)\left(-9\right)=36\\ \left(-4\right)+\left(-9\right)=-13\end{matrix}$
10

Rewrite the polynomial as the product of two binomials consisting of the sum of the variable and the found values

$-\left(x^2-13x+36\right)=0$
11

Factor the trinomial $\left(x^2-13x+36\right)$ finding two numbers that multiply to form $36$ and added form $-13$

$\begin{matrix}\left(-4\right)\left(-9\right)=36\\ \left(-4\right)+\left(-9\right)=-13\end{matrix}$
12

Rewrite the polynomial as the product of two binomials consisting of the sum of the variable and the found values

$-\left(x-4\right)\left(x-9\right)=0$
13

Multiply both sides of the equation by $-1$

$\left(x-4\right)\left(x-9\right)=0$
14

Break the equation in $2$ factors and set each factor equal to zero, to obtain simpler equations

$x-4=0,\:x-9=0$
15

Solve the equation ($1$)

$x-4=0$
16

We need to isolate the dependent variable $x$, we can do that by simultaneously subtracting $-4$ from both sides of the equation

$x=4$
17

Solve the equation ($2$)

$x-9=0$
18

We need to isolate the dependent variable $x$, we can do that by simultaneously subtracting $-9$ from both sides of the equation

$x=9$
19

Combining all solutions, the $2$ solutions of the equation are

$x=4,\:x=9$

Verify that the solutions obtained are valid in the initial equation

20

The valid solutions to the equation are the ones that, when replaced in the original equation, don't result in any square root of a negative number and make both sides of the equation equal to each other

$x=4$

Final answer to the exercise

$x=4$

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