Solve for y
y=5
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\sqrt{y-1}=y-3
Subtract 3 from both sides of the equation.
\left(\sqrt{y-1}\right)^{2}=\left(y-3\right)^{2}
Square both sides of the equation.
y-1=\left(y-3\right)^{2}
Calculate \sqrt{y-1} to the power of 2 and get y-1.
y-1=y^{2}-6y+9
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(y-3\right)^{2}.
y-1-y^{2}=-6y+9
Subtract y^{2} from both sides.
y-1-y^{2}+6y=9
Add 6y to both sides.
7y-1-y^{2}=9
Combine y and 6y to get 7y.
7y-1-y^{2}-9=0
Subtract 9 from both sides.
7y-10-y^{2}=0
Subtract 9 from -1 to get -10.
-y^{2}+7y-10=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=7 ab=-\left(-10\right)=10
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -y^{2}+ay+by-10. To find a and b, set up a system to be solved.
1,10 2,5
Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 10.
1+10=11 2+5=7
Calculate the sum for each pair.
a=5 b=2
The solution is the pair that gives sum 7.
\left(-y^{2}+5y\right)+\left(2y-10\right)
Rewrite -y^{2}+7y-10 as \left(-y^{2}+5y\right)+\left(2y-10\right).
-y\left(y-5\right)+2\left(y-5\right)
Factor out -y in the first and 2 in the second group.
\left(y-5\right)\left(-y+2\right)
Factor out common term y-5 by using distributive property.
y=5 y=2
To find equation solutions, solve y-5=0 and -y+2=0.
\sqrt{5-1}+3=5
Substitute 5 for y in the equation \sqrt{y-1}+3=y.
5=5
Simplify. The value y=5 satisfies the equation.
\sqrt{2-1}+3=2
Substitute 2 for y in the equation \sqrt{y-1}+3=y.
4=2
Simplify. The value y=2 does not satisfy the equation.
y=5
Equation \sqrt{y-1}=y-3 has a unique solution.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}