Final answer to the problem
$x=\frac{1}{2}+\sqrt{\left(\frac{5}{4}\right)},\:x=\frac{1}{2}-\sqrt{\left(\frac{5}{4}\right)}$
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Step-by-step Solution
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1
Move everything to the left hand side of the equation
$x^2+\left(x-1\right)^2-3=0$
Intermediate steps
2
Expand $\left(x-1\right)^2$
$x^2+x^2-2x+1-3=0$
Explain this step further
3
Subtract the values $1$ and $-3$
$-2+x^2+x^2-2x=0$
4
Combining like terms $x^2$ and $x^2$
$-2+2x^2-2x=0$
5
Use the complete the square method to factor the trinomial of the form $ax^2+bx+c$. Take common factor $a$ ($2$) to all terms
$2\left(x^2-x-1\right)=0$
6
Add and subtract $\displaystyle\left(\frac{b}{2a}\right)^2$
$2\left(x^2-x-1+\frac{1}{4}-\frac{1}{4}\right)=0$
7
Factor the perfect square trinomial $x^2+-xx+\frac{1}{4}$
$2\left(\left(x-\frac{1}{2}\right)^2-1-\frac{1}{4}\right)=0$
8
Subtract the values $-1$ and $-\frac{1}{4}$
$2\left(-\frac{5}{4}+\left(x-\frac{1}{2}\right)^2\right)=0$
9
Multiply $-1$ times $\frac{1}{2}$
$2\left(-\frac{5}{4}+\left(x-\frac{1}{2}\right)^2\right)=0$
10
Divide both sides of the equation by $2$
$\frac{2\left(-\frac{5}{4}+\left(x-\frac{1}{2}\right)^2\right)}{2}=\frac{0}{2}$
11
Simplifying the quotients
$-\frac{5}{4}+\left(x-\frac{1}{2}\right)^2=\frac{0}{2}$
$-\frac{5}{4}+\left(x-\frac{1}{2}\right)^2=0$
Intermediate steps
13
We need to isolate the dependent variable , we can do that by simultaneously subtracting $-\frac{5}{4}$ from both sides of the equation
$\left(x-\frac{1}{2}\right)^2=\frac{5}{4}$
Explain this step further
14
Removing the variable's exponent
$\sqrt{\left(x-\frac{1}{2}\right)^2}=\pm \sqrt{\left(\frac{5}{4}\right)}$
15
Cancel exponents $2$ and $\frac{1}{2}$
$x-\frac{1}{2}=\pm \sqrt{\left(\frac{5}{4}\right)}$
Intermediate steps
16
We need to isolate the dependent variable , we can do that by simultaneously subtracting $-\frac{1}{2}$ from both sides of the equation
$x=\pm \sqrt{\left(\frac{5}{4}\right)}+\frac{1}{2}$
Explain this step further
17
As in the equation we have the sign $\pm$, this produces two identical equations that differ in the sign of the term $\sqrt{\left(\frac{5}{4}\right)}$. We write and solve both equations, one taking the positive sign, and the other taking the negative sign
$x=\frac{1}{2}+\sqrt{\left(\frac{5}{4}\right)},\:x=\frac{1}{2}-\sqrt{\left(\frac{5}{4}\right)}$
18
Combining all solutions, the $2$ solutions of the equation are
$x=\frac{1}{2}+\sqrt{\left(\frac{5}{4}\right)},\:x=\frac{1}{2}-\sqrt{\left(\frac{5}{4}\right)}$
Final answer to the problem$x=\frac{1}{2}+\sqrt{\left(\frac{5}{4}\right)},\:x=\frac{1}{2}-\sqrt{\left(\frac{5}{4}\right)}$