Solve z^2=4z-3

$$z ^ { 2 } = 4 z - 3$$

Solve for z

$$z=1$$

$$z=3$$

Steps Using Factoring

Subtract 4z from both sides.

$$z^{2}-4z=-3$$

Add 3 to both sides.

$$z^{2}-4z+3=0$$

To solve the equation, factor z^{2}-4z+3 using formula z^{2}+\left(a+b\right)z+ab=\left(z+a\right)\left(z+b\right). To find a and b, set up a system to be solved.

$$a+b=-4 ab=3$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-3 b=-1$$

Rewrite factored expression \left(z+a\right)\left(z+b\right) using the obtained values.

$$\left(z-3\right)\left(z-1\right)$$

To find equation solutions, solve z-3=0 and z-1=0.

$$z=3 z=1$$

Steps Using Factoring By Grouping

Subtract 4z from both sides.

$$z^{2}-4z=-3$$

Add 3 to both sides.

$$z^{2}-4z+3=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as z^{2}+az+bz+3. To find a and b, set up a system to be solved.

$$a+b=-4 ab=1\times 3=3$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-3 b=-1$$

Rewrite z^{2}-4z+3 as \left(z^{2}-3z\right)+\left(-z+3\right).

$$\left(z^{2}-3z\right)+\left(-z+3\right)$$

Factor out z in the first and -1 in the second group.

$$z\left(z-3\right)-\left(z-3\right)$$

Factor out common term z-3 by using distributive property.

$$\left(z-3\right)\left(z-1\right)$$

To find equation solutions, solve z-3=0 and z-1=0.

$$z=3 z=1$$

Steps Using the Quadratic Formula

Subtract 4z from both sides.

$$z^{2}-4z=-3$$

Add 3 to both sides.

$$z^{2}-4z+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 3}}{2}$$

Square -4.

$$z=\frac{-\left(-4\right)±\sqrt{16-4\times 3}}{2}$$

Multiply -4 times 3.

$$z=\frac{-\left(-4\right)±\sqrt{16-12}}{2}$$

Add 16 to -12.

$$z=\frac{-\left(-4\right)±\sqrt{4}}{2}$$

Take the square root of 4.

$$z=\frac{-\left(-4\right)±2}{2}$$

The opposite of -4 is 4.

$$z=\frac{4±2}{2}$$

Now solve the equation z=\frac{4±2}{2} when ± is plus. Add 4 to 2.

$$z=\frac{6}{2}$$

Divide 6 by 2.

$$z=3$$

Now solve the equation z=\frac{4±2}{2} when ± is minus. Subtract 2 from 4.

$$z=\frac{2}{2}$$

Divide 2 by 2.

$$z=1$$

The equation is now solved.

$$z=3 z=1$$

Steps for Completing the Square

Subtract 4z from both sides.

$$z^{2}-4z=-3$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$z^{2}-4z+\left(-2\right)^{2}=-3+\left(-2\right)^{2}$$

Square -2.

$$z^{2}-4z+4=-3+4$$

Add -3 to 4.

$$z^{2}-4z+4=1$$

Factor z^{2}-4z+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(z-2\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z-2\right)^{2}}=\sqrt{1}$$

Simplify.

$$z-2=1 z-2=-1$$

Add 2 to both sides of the equation.

$$z=3 z=1$$