Solve x^2+12x-364

$$x ^ { 2 } + 12 x - 364$$

Factor

$$\left(x-14\right)\left(x+26\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-364. To find a and b, set up a system to be solved.

$$a+b=12 ab=1\left(-364\right)=-364$$

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 -364.

$$-1,364 -2,182 -4,91 -7,52 -13,28 -14,26$$

Calculate the sum for each pair.

$$-1+364=363 -2+182=180 -4+91=87 -7+52=45 -13+28=15 -14+26=12$$

The solution is the pair that gives sum 12.

$$a=-14 b=26$$

Rewrite x^{2}+12x-364 as \left(x^{2}-14x\right)+\left(26x-364\right).

$$\left(x^{2}-14x\right)+\left(26x-364\right)$$

Factor out x in the first and 26 in the second group.

$$x\left(x-14\right)+26\left(x-14\right)$$

Factor out common term x-14 by using distributive property.

$$\left(x-14\right)\left(x+26\right)$$

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.

$$x^{2}+12x-364=0$$

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.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-364\right)}}{2}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\left(-364\right)}}{2}$$

Multiply -4 times -364.

$$x=\frac{-12±\sqrt{144+1456}}{2}$$

Add 144 to 1456.

$$x=\frac{-12±\sqrt{1600}}{2}$$

Take the square root of 1600.

$$x=\frac{-12±40}{2}$$

Now solve the equation x=\frac{-12±40}{2} when ± is plus. Add -12 to 40.

$$x=\frac{28}{2}$$

Divide 28 by 2.

$$x=14$$

Now solve the equation x=\frac{-12±40}{2} when ± is minus. Subtract 40 from -12.

$$x=-\frac{52}{2}$$

Divide -52 by 2.

$$x=-26$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 14 for x_{1} and -26 for x_{2}.

$$x^{2}+12x-364=\left(x-14\right)\left(x-\left(-26\right)\right)$$

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

$$x^{2}+12x-364=\left(x-14\right)\left(x+26\right)$$

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.

$$x ^ 2 +12x -364 = 0$$

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

$$r + s = -12 rs = -364$$

Two numbers r and s sum up to -12 exactly when the average of the two numbers is \frac{1}{2}*-12 = -6. 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.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -6 - u s = -6 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -364

$$(-6 - u) (-6 + u) = -364$$

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

$$36 - u^2 = -364$$

Simplify the expression by subtracting 36 on both sides

$$-u^2 = -364-36 = -400$$

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

$$u^2 = 400 u = \pm\sqrt{400} = \pm 20$$

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

$$r =-6 - 20 = -26 s = -6 + 20 = 14$$

Evaluate

$$\left(x-14\right)\left(x+26\right)$$