Solve h^2+75h-6250

$$h ^ { 2 } + 75 h - 6250$$

Factor

$$\left(h-50\right)\left(h+125\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as h^{2}+ah+bh-6250. To find a and b, set up a system to be solved.

$$a+b=75 ab=1\left(-6250\right)=-6250$$

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 -6250.

$$-1,6250 -2,3125 -5,1250 -10,625 -25,250 -50,125$$

Calculate the sum for each pair.

$$-1+6250=6249 -2+3125=3123 -5+1250=1245 -10+625=615 -25+250=225 -50+125=75$$

The solution is the pair that gives sum 75.

$$a=-50 b=125$$

Rewrite h^{2}+75h-6250 as \left(h^{2}-50h\right)+\left(125h-6250\right).

$$\left(h^{2}-50h\right)+\left(125h-6250\right)$$

Factor out h in the first and 125 in the second group.

$$h\left(h-50\right)+125\left(h-50\right)$$

Factor out common term h-50 by using distributive property.

$$\left(h-50\right)\left(h+125\right)$$

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.

$$h^{2}+75h-6250=0$$

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.

$$h=\frac{-75±\sqrt{75^{2}-4\left(-6250\right)}}{2}$$

Square 75.

$$h=\frac{-75±\sqrt{5625-4\left(-6250\right)}}{2}$$

Multiply -4 times -6250.

$$h=\frac{-75±\sqrt{5625+25000}}{2}$$

Add 5625 to 25000.

$$h=\frac{-75±\sqrt{30625}}{2}$$

Take the square root of 30625.

$$h=\frac{-75±175}{2}$$

Now solve the equation h=\frac{-75±175}{2} when ± is plus. Add -75 to 175.

$$h=\frac{100}{2}$$

Divide 100 by 2.

$$h=50$$

Now solve the equation h=\frac{-75±175}{2} when ± is minus. Subtract 175 from -75.

$$h=-\frac{250}{2}$$

Divide -250 by 2.

$$h=-125$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 50 for x_{1} and -125 for x_{2}.

$$h^{2}+75h-6250=\left(h-50\right)\left(h-\left(-125\right)\right)$$

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

$$h^{2}+75h-6250=\left(h-50\right)\left(h+125\right)$$

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.

$$x ^ 2 +75x -6250 = 0$$

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

$$r + s = -75 rs = -6250$$

Two numbers r and s sum up to -75 exactly when the average of the two numbers is \frac{1}{2}*-75 = -\frac{75}{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.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -\frac{75}{2} - u s = -\frac{75}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -6250

$$(-\frac{75}{2} - u) (-\frac{75}{2} + u) = -6250$$

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

$$\frac{5625}{4} - u^2 = -6250$$

Simplify the expression by subtracting \frac{5625}{4} on both sides

$$-u^2 = -6250-\frac{5625}{4} = -\frac{30625}{4}$$

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

$$u^2 = \frac{30625}{4} u = \pm\sqrt{\frac{30625}{4}} = \pm \frac{175}{2}$$

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

$$r =-\frac{75}{2} - \frac{175}{2} = -125 s = -\frac{75}{2} + \frac{175}{2} = 50$$

Evaluate

$$\left(h-50\right)\left(h+125\right)$$