Solve x^2-3x-4

$${ x }^{ 2 } -3x-4$$

Factor

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

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-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 negative, the negative number has greater absolute value than the positive. 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.

$$a=-4 b=1$$

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

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

Factor out x in x^{2}-4x.

$$x\left(x-4\right)+x-4$$

Factor out common term x-4 by using distributive property.

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

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.

$$x^{2}-3x-4=0$$

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.

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

Square -3.

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

Multiply -4 times -4.

$$x=\frac{-\left(-3\right)±\sqrt{9+16}}{2}$$

Add 9 to 16.

$$x=\frac{-\left(-3\right)±\sqrt{25}}{2}$$

Take the square root of 25.

$$x=\frac{-\left(-3\right)±5}{2}$$

The opposite of -3 is 3.

$$x=\frac{3±5}{2}$$

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

$$x=\frac{8}{2}$$

Divide 8 by 2.

$$x=4$$

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

$$x=-\frac{2}{2}$$

Divide -2 by 2.

$$x=-1$$

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

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

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

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

Evaluate

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