Solve a^2+4a-77

$$a ^ { 2 } + 4 a - 77$$

Factor

$$\left(a-7\right)\left(a+11\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as a^{2}+pa+qa-77. To find p and q, set up a system to be solved.

$$p+q=4 pq=1\left(-77\right)=-77$$

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. List all such integer pairs that give product -77.

$$-1,77 -7,11$$

Calculate the sum for each pair.

$$-1+77=76 -7+11=4$$

The solution is the pair that gives sum 4.

$$p=-7 q=11$$

Rewrite a^{2}+4a-77 as \left(a^{2}-7a\right)+\left(11a-77\right).

$$\left(a^{2}-7a\right)+\left(11a-77\right)$$

Factor out a in the first and 11 in the second group.

$$a\left(a-7\right)+11\left(a-7\right)$$

Factor out common term a-7 by using distributive property.

$$\left(a-7\right)\left(a+11\right)$$

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.

$$a^{2}+4a-77=0$$

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.

$$a=\frac{-4±\sqrt{4^{2}-4\left(-77\right)}}{2}$$

Square 4.

$$a=\frac{-4±\sqrt{16-4\left(-77\right)}}{2}$$

Multiply -4 times -77.

$$a=\frac{-4±\sqrt{16+308}}{2}$$

Add 16 to 308.

$$a=\frac{-4±\sqrt{324}}{2}$$

Take the square root of 324.

$$a=\frac{-4±18}{2}$$

Now solve the equation a=\frac{-4±18}{2} when ± is plus. Add -4 to 18.

$$a=\frac{14}{2}$$

Divide 14 by 2.

$$a=7$$

Now solve the equation a=\frac{-4±18}{2} when ± is minus. Subtract 18 from -4.

$$a=-\frac{22}{2}$$

Divide -22 by 2.

$$a=-11$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 7 for x_{1} and -11 for x_{2}.

$$a^{2}+4a-77=\left(a-7\right)\left(a-\left(-11\right)\right)$$

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

$$a^{2}+4a-77=\left(a-7\right)\left(a+11\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.

$$x ^ 2 +4x -77 = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = -4 rs = -77$$

Two numbers r and s sum up to -4 exactly when the average of the two numbers is \frac{1}{2}*-4 = -2. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath-gzdabgg4ehffg0hf.b01.azurefd.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -2 - u s = -2 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -77

$$(-2 - u) (-2 + u) = -77$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$4 - u^2 = -77$$

Simplify the expression by subtracting 4 on both sides

$$-u^2 = -77-4 = -81$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = 81 u = \pm\sqrt{81} = \pm 9$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =-2 - 9 = -11 s = -2 + 9 = 7$$

Evaluate

$$\left(a-7\right)\left(a+11\right)$$