z^2+3z-4-এর সমাধান করুন

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

Factor

$$\left(z-1\right)\left(z+4\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as z^{2}+az+bz-4. To find a and b, set up a system to be solved.

$$a+b=3 ab=1\left(-4\right)=-4$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -4.

$$-1,4 -2,2$$

Calculate the sum for each pair.

$$-1+4=3 -2+2=0$$

The solution is the pair that gives sum 3.

$$a=-1 b=4$$

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

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

Factor out z in the first and 4 in the second group.

$$z\left(z-1\right)+4\left(z-1\right)$$

Factor out common term z-1 by using distributive property.

$$\left(z-1\right)\left(z+4\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

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

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

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

Square 3.

$$z=\frac{-3±\sqrt{9-4\left(-4\right)}}{2}$$

Multiply -4 times -4.

$$z=\frac{-3±\sqrt{9+16}}{2}$$

Add 9 to 16.

$$z=\frac{-3±\sqrt{25}}{2}$$

Take the square root of 25.

$$z=\frac{-3±5}{2}$$

Now solve the equation z=\frac{-3±5}{2} when ± is plus. Add -3 to 5.

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

Divide 2 by 2.

$$z=1$$

Now solve the equation z=\frac{-3±5}{2} when ± is minus. Subtract 5 from -3.

$$z=-\frac{8}{2}$$

Divide -8 by 2.

$$z=-4$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1 for x_{1} and -4 for x_{2}.

$$z^{2}+3z-4=\left(z-1\right)\left(z-\left(-4\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$z^{2}+3z-4=\left(z-1\right)\left(z+4\right)$$

Evaluate

$$\left(z-1\right)\left(z+4\right)$$