Solve 5h^2-24h-5

$$5 h ^ { 2 } - 24 h - 5$$

Factor

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

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 5h^{2}+ah+bh-5. To find a and b, set up a system to be solved.

$$a+b=-24 ab=5\left(-5\right)=-25$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -25.

$$1,-25 5,-5$$

Calculate the sum for each pair.

$$1-25=-24 5-5=0$$

The solution is the pair that gives sum -24.

$$a=-25 b=1$$

Rewrite 5h^{2}-24h-5 as \left(5h^{2}-25h\right)+\left(h-5\right).

$$\left(5h^{2}-25h\right)+\left(h-5\right)$$

Factor out 5h in 5h^{2}-25h.

$$5h\left(h-5\right)+h-5$$

Factor out common term h-5 by using distributive property.

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

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.

$$5h^{2}-24h-5=0$$

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.

$$h=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 5\left(-5\right)}}{2\times 5}$$

Square -24.

$$h=\frac{-\left(-24\right)±\sqrt{576-4\times 5\left(-5\right)}}{2\times 5}$$

Multiply -4 times 5.

$$h=\frac{-\left(-24\right)±\sqrt{576-20\left(-5\right)}}{2\times 5}$$

Multiply -20 times -5.

$$h=\frac{-\left(-24\right)±\sqrt{576+100}}{2\times 5}$$

Add 576 to 100.

$$h=\frac{-\left(-24\right)±\sqrt{676}}{2\times 5}$$

Take the square root of 676.

$$h=\frac{-\left(-24\right)±26}{2\times 5}$$

The opposite of -24 is 24.

$$h=\frac{24±26}{2\times 5}$$

Multiply 2 times 5.

$$h=\frac{24±26}{10}$$

Now solve the equation h=\frac{24±26}{10} when ± is plus. Add 24 to 26.

$$h=\frac{50}{10}$$

Divide 50 by 10.

$$h=5$$

Now solve the equation h=\frac{24±26}{10} when ± is minus. Subtract 26 from 24.

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

Reduce the fraction \frac{-2}{10} to lowest terms by extracting and canceling out 2.

$$h=-\frac{1}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 5 for x_{1} and -\frac{1}{5} for x_{2}.

$$5h^{2}-24h-5=5\left(h-5\right)\left(h-\left(-\frac{1}{5}\right)\right)$$

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

$$5h^{2}-24h-5=5\left(h-5\right)\left(h+\frac{1}{5}\right)$$

Add \frac{1}{5} to h by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$5h^{2}-24h-5=5\left(h-5\right)\times \frac{5h+1}{5}$$

Cancel out 5, the greatest common factor in 5 and 5.

$$5h^{2}-24h-5=\left(h-5\right)\left(5h+1\right)$$

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.This is achieved by dividing both sides of the equation by 5

$$x ^ 2 -\frac{24}{5}x -1 = 0$$

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

$$r + s = \frac{24}{5} rs = -1$$

Two numbers r and s sum up to \frac{24}{5} exactly when the average of the two numbers is \frac{1}{2}*\frac{24}{5} = \frac{12}{5}. 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>

$$r = \frac{12}{5} - u s = \frac{12}{5} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -1

$$(\frac{12}{5} - u) (\frac{12}{5} + u) = -1$$

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

$$\frac{144}{25} - u^2 = -1$$

Simplify the expression by subtracting \frac{144}{25} on both sides

$$-u^2 = -1-\frac{144}{25} = -\frac{169}{25}$$

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

$$u^2 = \frac{169}{25} u = \pm\sqrt{\frac{169}{25}} = \pm \frac{13}{5}$$

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

$$r =\frac{12}{5} - \frac{13}{5} = -0.200 s = \frac{12}{5} + \frac{13}{5} = 5$$

Evaluate

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