Solve k^2+36k-37

$$k ^ { 2 } + 36 k - 37$$

Factor

$$\left(k-1\right)\left(k+37\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as k^{2}+ak+bk-37. To find a and b, set up a system to be solved.

$$a+b=36 ab=1\left(-37\right)=-37$$

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. The only such pair is the system solution.

$$a=-1 b=37$$

Rewrite k^{2}+36k-37 as \left(k^{2}-k\right)+\left(37k-37\right).

$$\left(k^{2}-k\right)+\left(37k-37\right)$$

Factor out k in the first and 37 in the second group.

$$k\left(k-1\right)+37\left(k-1\right)$$

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

$$\left(k-1\right)\left(k+37\right)$$

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.

$$k^{2}+36k-37=0$$

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.

$$k=\frac{-36±\sqrt{36^{2}-4\left(-37\right)}}{2}$$

Square 36.

$$k=\frac{-36±\sqrt{1296-4\left(-37\right)}}{2}$$

Multiply -4 times -37.

$$k=\frac{-36±\sqrt{1296+148}}{2}$$

Add 1296 to 148.

$$k=\frac{-36±\sqrt{1444}}{2}$$

Take the square root of 1444.

$$k=\frac{-36±38}{2}$$

Now solve the equation k=\frac{-36±38}{2} when ± is plus. Add -36 to 38.

$$k=\frac{2}{2}$$

Divide 2 by 2.

$$k=1$$

Now solve the equation k=\frac{-36±38}{2} when ± is minus. Subtract 38 from -36.

$$k=-\frac{74}{2}$$

Divide -74 by 2.

$$k=-37$$

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 -37 for x_{2}.

$$k^{2}+36k-37=\left(k-1\right)\left(k-\left(-37\right)\right)$$

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

$$k^{2}+36k-37=\left(k-1\right)\left(k+37\right)$$

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.

$$x ^ 2 +36x -37 = 0$$

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

$$r + s = -36 rs = -37$$

Two numbers r and s sum up to -36 exactly when the average of the two numbers is \frac{1}{2}*-36 = -18. 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>

$$r = -18 - u s = -18 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -37

$$(-18 - u) (-18 + u) = -37$$

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

$$324 - u^2 = -37$$

Simplify the expression by subtracting 324 on both sides

$$-u^2 = -37-324 = -361$$

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

$$u^2 = 361 u = \pm\sqrt{361} = \pm 19$$

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

$$r =-18 - 19 = -37 s = -18 + 19 = 1$$

Evaluate

$$\left(k-1\right)\left(k+37\right)$$