Solve a^2+4a-5

$${ a }^{ 2 } +4a-5$$

Factor

$$\left(a-1\right)\left(a+5\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as a^{2}+pa+qa-5. To find p and q, set up a system to be solved.

$$p+q=4 pq=1\left(-5\right)=-5$$

Since pq is negative, p and q have the opposite signs. Since p+q is positive, the positive number has greater absolute value than the negative. The only such pair is the system solution.

$$p=-1 q=5$$

Rewrite a^{2}+4a-5 as \left(a^{2}-a\right)+\left(5a-5\right).

$$\left(a^{2}-a\right)+\left(5a-5\right)$$

Factor out a in the first and 5 in the second group.

$$a\left(a-1\right)+5\left(a-1\right)$$

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

$$\left(a-1\right)\left(a+5\right)$$

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.

$$a^{2}+4a-5=0$$

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.

$$a=\frac{-4±\sqrt{4^{2}-4\left(-5\right)}}{2}$$

Square 4.

$$a=\frac{-4±\sqrt{16-4\left(-5\right)}}{2}$$

Multiply -4 times -5.

$$a=\frac{-4±\sqrt{16+20}}{2}$$

Add 16 to 20.

$$a=\frac{-4±\sqrt{36}}{2}$$

Take the square root of 36.

$$a=\frac{-4±6}{2}$$

Now solve the equation a=\frac{-4±6}{2} when ± is plus. Add -4 to 6.

$$a=\frac{2}{2}$$

Divide 2 by 2.

$$a=1$$

Now solve the equation a=\frac{-4±6}{2} when ± is minus. Subtract 6 from -4.

$$a=-\frac{10}{2}$$

Divide -10 by 2.

$$a=-5$$

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 -5 for x_{2}.

$$a^{2}+4a-5=\left(a-1\right)\left(a-\left(-5\right)\right)$$

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

$$a^{2}+4a-5=\left(a-1\right)\left(a+5\right)$$

Evaluate

$$\left(a-1\right)\left(a+5\right)$$